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INTEGRATION 



BY 



TRIGONOMETRIC AND IMAGINARY 
SUBSTITUTION 

BY 

CHARLES O. GUNTHER, M.E. 

Assistant Professor of Mathematics and Mechanics in the 
Stevens Institute of Technology 

WITH AN INTRODUCTION 

BY 

J. BURKITT WEBB, C.E. 

Professor of Mathematics and Mechanics in the 
Stevens Institute of Technology 




NEW YORK 

D. VAN NOSTEAND COMPAITY 

23 Murray and 27 Warren^ Streets 

1907 



^" (/ 



jUBfi'ARYofCONGHEiiJ 
Two Copies HBC'jivu':^ 

I DEC Z im 

COPY B. 



Copyright, 1907, 

BY 

CHARLES O GUNTHER 

AND 

J. BURKITT WEBB 



• •• 



PREFACE. 



A METHOD of integration, which may be called the ^'Triangle 
Method/' has been used successfully for the past few years 
by the author in his classes. The primary object in view has 
been to eliminate the " Reduction Formulae/' and make the 
student independent of text^ 'bodlis and tables of integrals. 
This method is founded upon trigonometric principles with 
the result that the student gains proficiency not only in the 
integration of trigonometric differential expressions but also 
in the transformation of algebraic expressions into trigono- 
metric and exponential ones, and vice versa. 

This book is intended to be used in conjunction with the 
usual text-books on the subject, and should be taken up after 
the student has become familiar with the simple rules of 
integration, resulting from the reversion of the rules for 
differentiation. 

ChAS. 0. GUNTHER. 
Grand View-ox-Hudson, N. Y., 

January i, 190/. 



CONTENTS. 



INTRODUCTION. 

Page. 

Introduction 1 

Imaginary Quantities 2 

Exponentials 9 

Explanation of Figures and Tables 12 

Analytical Trigonometry 14 

Interrelations 16 

Integration 18 

Classification 21 



CHAPTER I. 

Trigonometric Differentials. 

Arts. 

1. Trigonometric Formulae 27 

2. Trigonometric Differentials 29 

3-8. Integration of t??/^coswi X sin^cccZx 30 

3. When eitlier m or n is an odd positive integer 30 

4. When m + n is an even negative integer 31 

6. When both m and n are even positive integers 32 

7. When 7?i ^ 71 is an odd negative integer 34 

8. When either m or n is an even positive integer, the other 

being a negative integer, and jn + n is a positive integer 

or zero 39 

9. Integration of d?/=cos mx cos nx dx, dy =sm mx cos nx dx, 

and c7?/ = sin mx ^in nx dx 41 

10-12. Integration of dy = enx cos hx dx, dy=€<^^ sin hx dx . . . . 42 

Miscellaneous Examples I 46 



VI CONTENTS. 

CHAPTER 11. 

Rationalization by Trigonometric Substitution. 
Arts. Page. 

13. Introduction 47 

14. Rationalization of Expressions containing v^a- - x- ... 47 

15. " " " '' \ a~ x^ ... 49 

16. ** " "• " Vx - a- ... 50 

17. Change of Form of Radical 52 

Miscellaneous Examples II 58 

18. Rationalization of Expressions Containing \/2ax — u:2 ... 59 

19. '' " " " V 2ax -r X- . . . eO 

20. *' " *' '•' trinomial surds . 62 

21. Binomial Differentials 63 

Answers to Examples Qb 



INTRODUCTION. 



Some time since I suggested to Professor Gunther that 
the instruction in Integral Calculus might be improved by 
teaching the student to integrate for himself a variety of 
expressions by the use of imaginary and trigonometric sub- 
stitutions, integration by parts and other methods, such as I 
use myself in the higher classes and which can easily be 
remembered, and giving up the use of tables of integrals and 
the customary formulj^ of reduction, especially the four 
standard formulae for raising and lowering the exponent, which 
are too cumbersome and difficult to remember and use. This 
will go far toward making the student independent of text- 
books in his integrations. 

Professor Gunther has shown such appreciation of these 
methods as to work them out with great care in a series of 
problems covering the ground of the ordinary text-book, leaving 
it to me to give a more general view of the principles in- 
volved as an introduction thereto. 

This introduction is therefore intended more for the teacher 
than the student, but it is placed here so that it may be in the 
hands of every student and thus reach those rare and thought- 
ful ones found in most classes who aim to master all the 
principles involved in a subject as well as the practical 
part. 

The most elementary acquaintance with as much of higher 
mathematics as is becoming necessary in the arts, requires 



2 INTRODUCTION. 

familiarity with imaginary and complex * quantities, which 
ought now to be included in elementary algebra, as well as 
with exponentials and with analytical trigonometry in 
connection with exponential quantities. Students come to the 
higher classes with much less practice in trigonometry and 
exponentials than they have had in algebra, and with none in 
imaginary quantities, and the practice of integrating all 
algebraic forms algebraically, and even transforming some of 
the trigonometric forms into algebraic, only increases the 
defect. It is therefore a great advantage to reverse the pro- 
cedure and give the student practice in analytic trigonometry, 
and other fundamentals of higher mathematics, in the inte- 
gration of algebraic forms by trigonometrical and other 
suitable methods. 



IMAGINARY QUANTITIES, 

Whether the nature and use of i = v — 1 should be made 
familiar to the student of ordinary algebra is scarcely a 
question, because there is no possibility of leaving it out ; and 
the only question is ; Shall he be made familiar with it as an 
impossible quantity, when its existence shows it to be possible, 
and shall it be called an imaginary quantity and no attempt 
be made to assist his imagination to grasp it, or shall expla- 
nations be given showing it to be a reasonable and useful 
quantity ? If in an exact science like algebra the logical 
development of the fundamental assumptions can lead to any- 
thing impossible, grave doubt is cast upon their correctness 
or the accuracy of the reasoning employed. 

* A complex quantity is the sum of a real quantity and an imaginary, 
i.e., it is a quantity part real and part imaginary. 



INTRODUCTION. 3 

The explanation required is not difficult. The square root 
of any minus number can be factored into the product of the 
square root of the number itself and \/ — 1, thus : V — 4 
= 2 \/ - 1, so that the only thing requiring explanation is 
\/ — 1. First then, if — 1 be recognized as a quantity, \/ — 1 
must also be a quantity, for it is inconceivable that the quanti- 
tative nature af — 1 can be destroyed by the operation of 
finding its square root when this operation has no such effect 
on other quantities ; or, if conceivable, — 1 must be a quantity 
of a different nature from others, the difference being respon- 
sible for this peculiar action of the square root upon it. 

What is, then, the nature of a square root, and what the 
difference between + 1 and — 1 ? 

The square root of a quantity is one of the two equal factors 
ivhose product is the quantity, thus : 

If h xh = a, h i^ the square root of a, consequently : 

// the square root of a quantity he used twice as a factor it 
produces the same result as using the quantity once as a factor, 

thus : cxbxb = cxa. 

To appreciate clearly the difference between + 1 and — 1 
it must be remembered that in algebra plus and minus are 
used in two ways, so that it is sometimes advantageous to 
distinguish them from each other. In the expressions m + ?^ 
and m — 7i the signs are supposed to represent addition and 
subtraction, and this is the first idea of them when they are 
used in arithmetic ; so that while 5 — 3 has a meaning, 3 — 5 
is held to be meaningless, for from a pile of 5 balls 3 can be 
taken away, but from a pile of 3 it is impossible to take 5. In 
algebra, however, a broader view is taken. 

In arithmetic four fundamental operations are taught, with 
a process for performing each, — addition, subtraction, multi- 



4 INTRODUCTION. 

plication, and division ; but in algebra these are reduced to 
two only, thus : if m = 5 and n = 3, arithmetic teaches how to 

add n to m, m + ?i = 5 + 3 = 8 . . , . {a) 

subtract n from vi^ vi — n = 5 — 3 = 2 . . . . (b) 

multiply m by n, 771 xn = 5x3 = 15 . . . (c) 

divide m hy n, m-r-n = 5-^3 = l^ . . . (d) 

But in algebra, if 7i is to be subtracted from m, no separate 
operation of subtraction is recognized, the rule being to 
change the sign of n and i^voceed as in addition ; and in the 
case of n added to m, symmetry of statement leads to, leave 
the sign unchanged and proceed as in addition, as the corre- 
sponding rule. 

There must, then, be a sign belonging to quantities, which 
can be changed or left unchanged, and this sign is called the 
intrinsic sign, which may be 4- or — ; besides which a second 
or copulative sign is needed to express the addition, and this is 
always plus. The algebraic expression of (a) Avill therefore be 

'^vi-}-'^n = m-\-n = 8 (a') 

and of (b) '^m -\-~n = m — n = 2 {h') 

where m + n and m — n may be regarded as abbreviated forms 
which generally lead to no ambiguity. 

In the same way algebra dispenses with division, for to 
divide vi by n it is sufficient to write ?i as a fraction and 
multiply, thus : 

m X — = — = 1|. 
n n 

A much closer parallel appears, however, in the customary 
algebraic use of exponents. An exponent is a small numeral 
at the upper right-hand corner of a quantity which states two 



INTRODUCTION. 5 

things, first, that the quantity or number is to be used as a 
factor, and second, the number of times it is to be so used. 
According to this usage (c) may be written algebraically 

m^Xn^ =vi.n =vin =15 .... (c') 
and {d ) m^x n-^ = in . n-'^ = mn-^ ==^3 • . • (^0 

for the rule is to change the sign of the exponent of n and pro- 
ceed as in niidtiplicaiion, whereas in {c') we leave the sign 
unchanged and proceed as in midtip)lication. The forms 
m . n, 77in, etc., may be regarded as abbreviations, which gener- 
ally lead to no ambiguity. Here instead of the copulative 
sign of addition there is that of multiplication, x , or the dot 
used in algebra, and instead of the intrinsic signs of the 
quantities there are the signs of the exponents. 

The failure to keep in mind and distinguish between the 
copulative and intrinsic signs in addition is a source of con- 
fusion in many problems, as, for instance, in logarithms when 
the intrinsic sign of the characteristic is different from that 
of the mantissa, and it would be advantageous if separate 
symbols were provided for them. 

It would be better if the copulative sign for addition 
were — , so that in — n should mean ni plus n. It is the simpler 
mark of the two, and m — n could be changed to m + tz by 
adding a stroke ; whereas with the signs as they now are, the 
change from m plus n to m minus n requires an erasure, and 
this change has to be made much oftener than the contrary 
one. 

It would also be better in algebraic multiplication, Avhich 
includes division, to use the line already employed to indicate 
a fraction, or, in other words, the line employed to indicate 

division, for the general sign of multiplication, thus ; — means 

n 



6 INTRODUCTION. 

VI divided by n, or m multiplied by — . Why not write mn or 

n — 

mn for m multiplied by n, so that while a line in front of a 

quantity would indicate that it was to be added, as in di - n 

{n added to m), a line under it would mean that it was a 

factor ? The algebraic rule dispensing with division and 

reducing it to multiplication would then be similar to that for 

subtraction and would read : To divide by a quantity, change 

the factor al sign from under to over it and loroceed as in 

midtiplication^ thus : m and n are both factors, and mn are 

two factors multiplied together ; but if m is to be a multiplier 

and n a divisor, then they must be written m and ~n^ and when 

multiplied together would appear as 7nn ox — OYmln, (c) would 

— n 

then be written thus : 

mM = 5x3 = 15, 

and (d) thus : mn = — =5-^3= — =lf. 



Now as to the difference between + 1 and — 1, these signs 
are evidently intrinsic signs, and the — 1 is not to be thought 
of as 1 subtracted from anything. What, then, do these 
intrinsic signs indicate ? It will be found upon consideration 
of this question that the intrinsic sign of some physical 
quantities is by nature plus and cannot be minus. A cubic 
foot of water has about two units of mass in it, and the same 
volume of a substance of half that density would have one 
unit, and, further, in a cubic foot of vacuum there would be 
zero units of mass ; but to have - 1 or - 2 units of mass in a 
cubic foot is inconceivable, and may well be said to be impos- 
sible, so that the intrinsic sign of density is plus. Also the 
intrinsic sign of the radius of a sphere or a circle is + , and 



INTRODUCTION. 7 

the device of a minus value to the radius is only such, and 
sometimes leads to error. There are, however, quantities to 
which nature allows both plus and minus intrinsic signs. 

In rectangular co-ordinates the intrinsic sign of either x or ij 
may be plus or minus, + meaning a distance measured in an 
assumed direction, say to the right of the origin, and — mean- 
ing in the opposite or left-hand direction, and in this way a 
rational interpretation of -f 1 and — 1 is obtained as two unit 
distances measicred in opposite directions from an origin or zero 
point. They may therefore be properly represented by arrows 
of unit length drawn to the right and left, thus : 

- 1 < O > 4- 1 

an arrow (or directed line, or vector) having two properties 
and no more, magnitude or length, and direction, the same 
as quantities + 1 and — 1 have. 

4- 1 and — 1 may then be regarded as referring not only to 
the unit distances themselves, but to the points at the ends of 
the arrows. 

^ow - 1 or + 1 regarded as a factor must be somewhat 
different from the — 1 and + 1 regarded as points or distances 
from the origin. Indeed, in the ordinary multiplication of 
5 by 3, 5 is termed the multiplicand and 3 the multiplier, and 
3 is supposed to operate as a factor on 5, 5 being trebled by 
the operation and not regarded as a factor, so that a difference 
is recognized, which can be further emphasized by considering 
that there would be no difficulty in multiplying — 1 by 3, but 
only in the reverse operation. To get out of the difficulty it 
might have to be assumed that the reversal should make no 
difference in the result, but this would not help the matter 
if — 1 were to be multiplied by — 3. 

But in algebra factors can be minus as well as plus, and a 
remarkable rule is given for their multiplication. The num- 



8 INTRODUCTION. 

bers or quantities themselves are to be multiplied in the usual 
way for any number of factors, d^s '^m .~n .~p . ~q, and a sign 
is to be prefixed to the product, + if the number of 7ninus signs 
is even, and - if it is odd^ and this is an arbitrary rule for 
which no justification is given except what it gets subsequently 
from the fact that it works well. 

Now this rule of sign might equally well have been arbitra- 
rily stated in either of the following forms : 

Call a plus sign zero and a minus sign one, and add together 
the signs of the factors ; if the result is an even nurnber put + ; 
and if an odd number pat — before the product ; or, supposing 
angular measure to be understood : 

Call a plus sign 0^ and a minus sign 1%^^, and add together 
the signs of the factors ; if the result is a multip>le of 360° the 
product is plus, and if not it is minus. 

It must then be evident that the rule given is equivalent to 
a statement that the effect of the factor - 1 is to reverse the 
direction of the quantity on which it operates without altering 
its magnitude, which can only be done by revolving it about 
the origin through 180°. 

Therefore the use of —1 as a factor does not alter its 
compound nature ; it is still a directed quantity, and as such 
may properly be represented by a line drawn to the left of 
the origin ; but instead of the minus sign meaning simply the 
direction to the left, it now represents a revolution of 180°, 
and its operation as a factor is to revolve another quantity 
(considered as a multiplicand) through 180°, thus changing it 
from whatever direction it may have to the opposite one. 

The meaning of the square root of minus one and its effect 
follow directly from the foregoing and the definition of square 
root. The quantity V - 1 used twice as a factor must be 
equivalent to — 1 used once, and cause a revolution of 180° ; 
it must therefore, when used once, cause a revolution of 90^, 



INTRODUCTION. 9 

Starting then with the directed line or arrow -f- 1 as the 
multiplicand, 



V - i.V ~ 1= -1, 



-l.\/-l=-\/-l, 



and -V - 1 . V - 1 = 1. 



As a directed quantity \/ — 1 is therefore unity measured in 
the direction of the Y-axis, and as a factor it is a revolution 
tln^ough a right angle in that angular direction which is con- 
sidered positive. I is used to represent the plus square root 
of minus one, or i = \^ — 1, and is called the wiaginary unit 
to distinguish it from + 1 which is called the real ttnit, 
although one is just as real as the other, differing in direction 
only, as do the co-ordinates x and y. i = \^ — 1 is therefore 
an actual algebraic quantity as easy to use after a little 
practice as any other, and very useful in many cases. It 
furnishes an elegant algebraic method of working with x and y 
co-ordinates, by means of complex quantities, without having 
to keep them separate from each other, and includes also 
polar co-ordinates by the help of exponentials. 

All points in a plane have therefore their algebraic repre- 
sentation as shown in the following figures. See page 10. 

EXPONENTIALS. 

Familiarity with exponential quantities is important in 
mathematical work of all kinds. The base e of the Natural 
System of Logarithms was not invented, any more than 
V — i was, to plague the brains of unwilling students, but 
it is a Constant of Nature, the same as tt is, and was discov- 
ered, as any rare mineral might be unearthed, by mathe- 
matical digging. Like a rare mineral, a constant of nature 



10 



INTRODUCTION. 






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INTRODUCTION. 



11 



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12 INTRODUCTION. 

EXPLANATION OF FIGURES AND TABLE. 

Figures 2 to 6 differ only in the values ascribed to the points 
shown. These values could not easily be written upon the 
figui-es, so that instead of drawing five separate figures, each 
with its set of values, but one is shown with reference letters at 
the points. These letters are placed in the first column of the 
Table, headed ^^ Points/' and in the following columns the values 
belonging to the points are given for each of the five figures. 

In Figure 2 each point, or the vector from the origin to the 
point is a power of a real number, thus ; the origin zero is , 
x-"^ = where x is any real number, and point 1 is unity to 
any power x or x^ ; a, b, etc., are given both as powers of + 1 
and — 1 and the points on the spiral as powers of — 2. Other 
values also belong to the points, thus ; one of the four fourth 
roots of — 1 is a, the other three are d, e and h. The three 
cube roots of — 1 are ^, - 1 and g, and the two square roots i 
and —i. The four fourth roots of + 1 are l,i, — 1 and — i, 
the three cube roots are 1, c and /, and the two square roots 
are 1 and — 1. 

In Figure 3 each point is a power of the Imaginary Unit i 
or a multiple thereof, so that the points can be represented by 
powers of either real or imaginary numbers at will. 

In Figure 4 each point is regarded as a Complex Quantity z, 
given in the Standard Algebraic Form z = x -]- iy, x and y 
being its Cartesian Coordinates, thus ; h has x == 2~^ and y = 3~i 

In Figure 5 the Trigonometric Form z = p (cosine x^ i sine x) 
is used, thus ; n has p = 1.414 and x = Itt. The symbols 
used in this column stand, as might be inferred, for the words 
*^ cosine " and " sine," the use of words or abbreviations thereof 
in a formula being regarded as inelegant and wasteful of room. 

In Figure 6 the Exponential Form ^ = e^ is given. To 
save room x only is tabulated. 



INTRODUCTION. 13 

possesses wonderful and valuable properties, which are dis- 
covered upon examination and analysis. This constant e is 
represented by one of the simplest of series, for, as shown by 
the Differential Calculus, 

/v»2 /-jfO /y»4 

e"^ = x^ + x^ -\ — - H H 4- ... etc., 

|2 ^ L£ 

and therefore, 

,^,i = l + l+_ +-+-+ . . . etc., 

each term being divided by its number to form the next term. 

A more comprehensive idea of e-^ must be taken than that 
given in connection with ordinary logarithmic and exponen- 
tial work. It is there taught that a minus quantity has no 
logarithm, which may seem true from the fact that the real 
numbers from — go to + go are all needed as logarithms for 
the plus numbers, integral and fractional, between and + oo . 
Where, then, can logarithms for minus numbers be found ? 

In the series for ^^ if x be made imaginary, say x = 16, 
6 being real, c-^ = e'^ as will be seen later, will always be a point 
on the unit circle (or circle of unit radius) having the origin 
for its center, and in the same way re'^ is a point on the circle 
of radius r, also it will be found that 

ei^= - 1, or, 

iV= logarithm of — 1; further, 

rci^ = ^logr + 17T ^ _ ^,^ therefore 
log r 4- ^7r=log ( - r), 

so that the logarithm of a minus number is the logarithm of 
the same plus number plus iir. 

There is, then, an important connection betw^een these two 
constants of nature, e and tt, and other numbers, e has also 
other important trigonometrical connections, especially in 



14 INTRODUCTION. 

connection with i, i being in fact the foundation quantity in a 
:Sort of trigonometry in which angles are represented and 
treated algebraically along with distances, as may be inferred 
from the last two figures. This makes it necessary to gen- 
eralize and extend the principles of trigonometry. 

ANAL YTICAL TRIG ONOME TR Y, 

The two fundamental functions in analytical trigonometry 
are the cosine and its complementary function the sine, and 
the Differential Calculus expresses them both as series of 
powers of the angle x. But the idea of these quantities 
obtained in ordinary trigonometry, as the ratios between the 
sides of a triangle, must be generalized for them to be of use 
in higher mathematics ; and to do this we drop the old defini- 
tions and define these functions of x anew so as to generalize 
and extend their meanings. The cosine and sine are there- 
fore defined as the sums of their series of powers of x-, thus : 



x' 



y,Q 



Gosmex = x^- 1 h . . . etc., 

11 [± [1 

/y>3 /y»5 ^7 

T • -I lAy tAj %Aj I 

and sine x = x^ — v — - + . . . etc. 

13 15 7 



The cosine is written first, contrary to the usual custom, 
because it is the simpler and more important function, and 
corresponds with the x direction or real axis^ while the sine 
corresponds with the y direction or imaginary axis. 

These definitions are a generalization of the functions and 
include the old definitions. In this connection the nature of 
X must be understood and a false idea of it corrected, x is 
usually regarded as an angle, but in reality so to regard it is 
nothing more than to remember that it refers to an angle in 



INTRODUCTION. 15 

the particular problem concerned, for mathematically x cannot 
be anything more than the number expressing the angle in 
radians. The generalization of the cosine and sine therefore 
consists in calling the sums of the above series by the same 
names, ^^ cosine of x '' and "sine of x^^ even when cc is a number 
obtained by measuring some quantity other than an angle. 

This general view allows us also to extend the meaning of 
the functions to include Imaginary and complex values of cr, 
and, without trying to form any idea as to what an imaginary 
or complex angle could or might be, it is allowable for con- 
venience to still call X " the angle x '^ when it is not an angle, 
and "the imaginary angle x^'^ when it is an imaginary or 
complex number. 

In ordinary trigonometry it is taught that cosines and 
sines all lie between + 1 and - 1, so that a cosine equal to 2 
would be an impossibility, but by supposing x to be imaginary 
it becomes possible, and cosines and sines can have all values 
from 4- 1 to + 00 ; also by giving x a suitable complex value 
these functions take values between — 1 and — oc , other 
complex values for x giving complex values for the functions, 
so that any point in the co-ordinate plane, real, imaginary, or 
complex, can be represented in analytical trigonometry by the^ 
cosine or sine of some x^ but this x can only represent a real 
angle in the case that the functions lie between + 1 and — 1.. 

The effect of an imaginary value of x on the cosine and sine 
and on s^ will appear by placing x = lO in the series and 
simplifying the results ; thus it will be found that 

en a\ n3 

cosine a:; = COS t^ = 1 -r r^r + ,-7-+ ,-^ + • • • ^^^-y ^^^ 

2 4 5 

/ B^ B^ B^ \ 

sine x = ^miB=i{6 -\' ,-^ + —r+ r^ + • • • ^^c. ), 

i_l_ l__ lL ^ 



16 INTRODUCTION. 

to which may be added 

Cfl Z)3 Z34 /)5 

^x^^^B^\^iQ_ ^ _/ Z_ + _?_ ^i^-- . . . etc. 

12 I 3 I 4 I 5 

for purposes of comparison. 

The first thing to be noticed is that the cosine and sine 
series are simpler for imaginary angles than for real, and 
special names have been given to these simpler series, the first 
being called the hyperbolic cosine of 6, and the second (without 
the i in front of it) the liyperholic sine of 6, which leads to two 
equations, which may be taken equally well for definitions of 
the hyperbolic functions, 

hyp. cosine = cosine 10, 
hyp. sine 0= —i sine id. 

Hyperbolic cosines and sines lead to a complete hyperbolic 
trigonometry parallel with ordinary trigonometry but having 
differences which perplex the memory. It is therefore an 
additional advantage in the use of imaginary quantities that 
their use amounts to including hyperbolic trigonometry in 
common trigonometry, which is much simpler than learning 
and using it separately. Had trigonometry been developed 
from the analytical standpoint, it is likely that it might have 
had as its basis the hyperbolic cosine and sine as simpler than 
the circular. 



INTEERELA TIONS. 

Having, now these generalized and extended views of the 
fundamental quantities ^, £% cosine ;r, and sine .t, their rela- 
tions to each other must be further examined, being of the 
greatest practical use in higher mathematics. 



INTRODUCTION. 17 

Eearranging the series last given, there results 



fP fl* ff^ 
£*« = !-— 4- —-— +... etc. 

2 14 16 



... e^ 9' 0' 

-,^^e- _+_-,-^+ ...etc. 



Y 



which being compared with the defining series for cosine and 

sine gives 

s^^ = cosine 6-^i sine Oy 

and in the same way it is found that 

Q-id = cosine 6-i sine 6, 

which also comes directly from the previous formula, because 
changing the algebraic sign of an angle changes the algebraic 
sign of its sine. 

These values for e^^ and e-*^ are complex quantities and 
may be laid off as shown in Figure 5 and Figure 6. The cosine 
being real is to be laid off as an x co-ordinate from the origin 
along the X= axis, and at its end the imaginary quantity i sine 
is to be laid off as the y co-ordinate, plus in one case and 
minus in the other. Evidently then £*^, as previously stated, 
will always be on the unit circle, and to find a point repre- 
senting rs^^ is simply to increase the radius from unity to r, so 
that any point in the plane can be expressed by r£^^ or e'^^s r+ie 
as shown in the foregoing figures. This amounts to a sort of 
exponential trigonometry. 

This representation of any point in the co-ordinate plane by 
exponentials leads to a conception either of the point itself, 
or of the radius from the origin to the point as a single 
quantity, which may be represented by a single letter, thus. 



18 INTRODUCTION. 

z= £^osri-i9 = ^*^ie^ 2^;Yid this conception is justified by the fact 
that this z can be put into an equation and subjected to 
algebraic treatment like any other quantity without error. 

By addition and subtraction of the last formulae exponen- 
tial values are arrived at for cosine and sine which are highly 
important, and should be memorized. 

cos(9=^-ti — 

2 

sin (9=^-—? — , 
the corresponding formulae for hyperbolic functions being 
hyp. cos = 



2 ' 



pQ _ ^- 

hyp. sin 6 = =- 



■ J 



which again emphasizes the greater simplicity of these 
functions. 

From the same two formulae are obtained 

log (cos -hi sin 6) = iO, 
log (cos 6 — i sin 6) = — i6. 

In the following pages it will be shown how these principles 
of higher mathematics apply to simple integration. 

INTEGRA TION. 

The integration of algebraic forms containing the radical 
\/± (^2 4- 2ax + h), a and h having any values whatever, 
depends on the removal of the radical sign, which can be done 



INTRODUCTION. 19 

by imaginary and trigonometric substitution. As physical 
problems often depend upon angles, the change to a trigono- 
metric form is likely to simplify the problem, and often in a 
marked degree, besides furnishing interesting and valuable 
practice in the principles developed. Thus, 

putting x = u — a the radical reduces to 



\/v? ±_ c^ where c^ = ^ — a^ is ± 

according to the sign and magnitude of h. Both signs can 
now be changed together if desired, by multiplying the radical 
by i and dividing inside by - 1, which gives 



and then the substitution u =w will change the sign of u^ and 
give, if desired, 

i\/ + ^^ =F c^ 

so that the original radical can be reduced to any one of three 
forms, 

\/x^-{-c^, \/x^ — c^, or- \/c'^ — x^y 

and these radicals can be removed by substituting x = c tan 9 
in the first, x = g sec 6 in the second, and x = g cos or c sin 
in the third. By the imaginary factor therefore the signs 
can be changed at will so as to make use of the particular 
trigonometric substitution desired ; and the fact that the 
differential coefficients of cosine and sine are of the first 
power, and those of the tangent and secant of the second 
power, m-akes it possible by this choice to change the 
exponents in the resulting form. 



20 INTRODUCTION. 

If the original expression is or leads to one of the simpler 
forms, as 

d 



J dx 



(f) 



where the quantities may be real or imaginary, there is no 
necessity for trigonometric substitution, though it is interesting 
to make such for purpose of comparison, and the integration 
can be made directly into a circular function, and thereby 
practice be had in the use of imaginary arcs and with the 
exponential values of cosine and sine, and with constants of 
integration. 

Much can be learned by integrating the same expression 
in various ways ; the result may appear in different forms, and 
it is excellent practice to reduce one form to another, which 
can always be done if integrated correctly. In such a reduc- 
tion the constant of integration plays an important part. 

It remains then to consider the integration of simple trigo- 
nometric forms, which will be classified and discussed in a 
general but orderly manner. It is not intended that teacher 
or student shall follow this classification ; indeed, it is better 
to avoid all classification, as more will be learned by solving 
each problem independently according to the general princi- 
ples given. Many problems can be solved by several methods, 
and that given in the classification may not always be the 
simplest. Then again a different classification might be 
made, and more is learned by making or attempting one than 
by following one already made. 

But this classification will nevertheless be of use, for tlie 
fact of having one that covers the whole ground (those in text- 
books do not always do so) is proof that all the forms can be 



INTRODUCTION. 21 

integrated by the principles given ; and further, if some of 
them are attempted without success, the classification can 
then be consulted. Of course, without any classification it 
can be seen that all such forms may be integrated, for the 
exponential values of cosine and sine may be substituted, 
when the expression will simplify into a series of exponential 
terms of various powers which integrate at once, but this is 
not an easy way for many problems. 

CLASSIFICATION, 

Any product of integral powers of simple trigonometric 
functions can be put in the form 

cos^ X sin^ X dxy m, an integer 5 ^j an integer. 

The possibility of cases appearing in the analysis where the 
result depends on whether vi > or < n might suggest a pro- 
vision against such ambiguity' by supposing at once m > n. 
This would necessitate an equal number of cases of the comple- 
mentavy ]oTm^ 

sin^ X cos^ X dxj 

which is equivalent to m < 7^ in the original form. 

However, this complementary form may be disposed of 
as follows : 

Put {x-\-y) = — , then cos x = sin y, sin x = cos y^ dx= — dy^ 

and we get, sin^ x cos^ x dx= ~ cos^ y sin^ y dy, 

which reduces the complementary to the standard form with 
the sign changed ; so that the integral of the complementary 
form can be obtained by integrating the corresponding standard 
form and changing the sign. If the integral be obtained in 
this and some other way, the constants of integration may need 
adjustment to make the results agree. 



22 



INTRODUCTION. 



The standard form, 
cos^ X sin^^ X dx, 



771, integral and 5 ^? integral, 



therefore, includes all forms, which must belong to one of the 
sixteen cases symmetrically arranged in four groups in the 
following table : 









m 




+ 


- 


n 


+ 




odd 


even 


odd 


even 


odd 


a 


a 


a 


a 


even 


a 


c 


d 


c 


- 


odd 


a 


d 


b 


d 


even 


a 


c 


d 


h 



The groups a, hy c and d will now be considered separately. 

Group a. 

One exponent + odd, the other ± odd or ± even. 

cos^ X sin^ X dx = sin^ x cos^-^ x cos x dx^ 
if m is + odd and = 2p + 1, 

= sin^ x {1— sin2 xY d (sin x) 

= (sin^ X — j9 sin^+2 x + q sin^+* x— . . . etc.) d (sin x) 

where the terms of the series integrate as powers of sin x ; 

or ; cos^ x sin^ x dx = cos^ x sin^^ x sin x dx, 
if 7^ is + odd and =2p + l, 

= — cos^ X {1 — cos^ xy d (cos x) 

= — (cos^ X —p cos^'+'^ x + q cos^+^ X— .... etc.) d (cos x)^ 

where the terms integrate as powers of the cosine. This second 
supposition is equivalent to the complementary form. 



IN^IRODUCTION. 23 

Group b. 

Both exponents — odd or both — even. 

For convenience put the sum of the exponents = - 2^, we 
shall now always have 2p + even. 



, sin^ X cos^^ X cos^ x dx 
cos^ x sm^ x dx= ^ 

cos^ x 

= tan^ X cos^+^ x dx = tan^ x sec^^"^ ^ ^qq2 ^ ^^ 

= tan^* X {!-{- tan2 :;c)^~"^ 6^ (tan x) 

= {tan^ cc + (i? - 1) tan^+2 ^^_|_ ^ tan^+^ x^ etc.j d (tan :;c), 

the terms of which integrate as powers of the tangent. These 
powers start as minus powers, but may become plus toward 
the end of the series. It evidently makes no difference 
whether m ^ n. 

Group c. 

Both exponents even, one + even and the other ± even. 
Let the exponent which is plus even = 2p, 

cos*^ X sin^ xdx = cos^ x (1 — cos^ x)^ dx, \i n = 2pj 
= (cos^ X —p cos^+^ x-\-q cos^+^ X, etc.) dx, 

or ; = sin^ x (1 — sin^ x)^ dx, if m = 2p, 

= (sin^ x—p sin^+2 x + q sin^+^ x, etc.) dx. 

If both m and n are plus, the exponents of the terms of the 
series will be plus, but if one is minus there will be minus 
exponents in the series. All the exponents will, however, be 
even. 

Terms with plus exponents integrate by doubling of 



24 INTRODUCTION. 

the angle, as often as may be necessary, by means of the 

formulae 

1 + cos 2x . , 1 - cos 2x 
cos- X = ^ , sm^ X = . 

Terms Avith minus exponents fall under group h by letting 
the other exponent equal 0. 

When m + ^^ is - even, the method of group h can also be 
employed. 

Group d. 

One exponent — odd, the other ± even. 

By the use of one of the following imaginary trigonometric 
substitutions the — odd exponent can be changed to plus. 

Put sin x = i tan and there results the following set of 
values : 

cos X = sec ^, sec x = cos 0, 

sin x= I tan 0, esc x= —i cot 6, 

tan X = i sin 0, cot x= —i esc 6; 

dx = i sec (W, 

and for cos x = I cot 6 there result the complementary 
substitutions : 

sin X = CSC 6, esc x = sin 0, 

cos x= i cot 6, sec x= — i tan 6, 

cot X = i cos 9, tan x= —i sec 6, 

dx = I QSG dOj 

The first gives 

cos^^ x sin^ X dx = sec^ {i tan ^)^ ^ sec d0 
= ± i cos-(^+'^+i)^ sin^ ^ cif^, 

and the second gives 

cos^ X sin^ X dx^ (I cot 0)"^ cso^ ^ i esc d0 
= ± i oos^^ sin-^"^+^^+i)(9 c^^. 



INTRODUCTION. 25 

As m 4- ^ 4- 1 is always even, the first should be used when m 
is odd, and the second when n is odd. Both exponents are 
now even, and, as can easily be seen, one will always be plus, 
which places them in group c. 



The following use of the Imaginary is interesting on account 
of the mechanical analogy involved. 

To integrate du = e^ cos x dx; 

add to it the complementary equation 

dv = e^ sin x dx 
multiplied by the Imaginary Unit i and get 

du + i dv = £^ (cos X + i sin x) dx, or 

d (u + iv) = dz = £'^ £'^ dx = e^^-^'^'^dx. 
dz is here a Complex Quantity, but in integrating it the real 
and imaginary parts must integrate separately and form the 
real and imaginary parts of the integral. The result is 
therefore 

z = u + iv = —^ £(1+^)^4- C 

1 + ^ 

1-i 

= —^ £^ (cos x-hi sin x) + C 

Separating the real and imaginary parts 

-u = i e^ (cos X + sin x) + C^ 
IV = \ e^i (sin x - cos x) + C2, or 

v=^ e^ (sin x - cos x) + Cg . 

The given equation represents a periodic motion along the 
Keal Axis and the equation added represents one along the 
Imaginary Axis of an equal and similar degree of complexity, 



26 INTRODUCTION. 

but the combined equation represents a simple spiral motion 
around the origin with the variable radius e^, with an equally 
simple integration. The simplest form of this mechanical 
analogy is the combination of two mutually rectangular 
harmonic motions to form a uniform motion in a circle. 

Or let dPu = e^ sin x dx^ 

and add icPv = ie^ cos x dx- ; 

then d'^-z = d'^ (u + iv) = ie^ (cos x - i sin x) dx^ 

== is' '-'''' dx\ 

Integrating, there results 

1-V 
i 

Integrating again 



-dz={e'-'-''^+C)dx. 



i 

where C^ = C (l-i), 

or -2z= -2 (u + iv) = £^ (cos x-i sin x) -\-C^x+C^. 

Separation of the real and imaginary parts leads to 

- 2ii = e^ cos X + C3 X + C4, 
2v = r"^ sin x + C5 x + C^. 

It may readily be inferred that complex quantities are as 

important and necessary in mathematical as revolving bodies 

are in mechanical work and analogies between them are often 

helpful. 

J. BURKITT WEBB. 

HOBOKEN, N. J., 

January /, 1907, 



IXTEGRATION BY TRIGONOMETRIC 
AND IMAGINARY SUBSTITUTION. 



CHAPTEE L 

TRIGONOMETRIC DIFFERENTIALS. 

1 . Trigonometric Formulae . — The method of integration 
developed in the following pages is designed to replace the 
usual '' reduction formulae/' and being founded upon trigo- 
nometry the more important trigonometric relations are, for 
convenience of reference, tabulated below. 

i. cos X = , sm X = , tan x = 



sec X esc X cot x 

o . sin X , cos X 

2. tan X = , cot x = — — . 

cos X sin x 

3. cos I ^ - cc J = sin a:!, 4. cos^ a: + sin^ a: = 1, 
sin ( — —x\ = (iOB>x, sec^ 0^ — tan- a- = 1, 

tan ( — — x\ = cot x csc^ x - cot^ x = l. 



5. cos {x ±^ y) = cos x cos y =F sin x sin y, 

6. sin {x ± y) = sin x cos y ± cos x sin y. 

r^ , , , . tan X 4- tan ?/ 

i . tan {x ± y) = — ^ . 

1 =f tan X tan y 

8. cos 2x = cos2 X - sin^ a: = 2 cos^ a^ - 1 = 1 - 2 sin- a: 

27 



28 INTEGRATION. 

2 tan X 



9. si a 2x = 2 sin x cos a:-, tan 2x -- 



1 — tan^ X 



^ ^ , 1 + cos 2x . ^ 1 - cos 2.7; 

10. cos^ X = , sm^ X = . 

2 2 

11. cos a:? + cos y= 2 cos ^ (x-^y) cos ^ (-^'-Z/)- 

12. cos a:* — cos y= —2 sin-i {x + y) sin J {x — y). 

13. sin cT + sin ?/= 2 sin ^ {x + y) cos ^ {x — y). 

14. sin X' — sin ?/ = 2 cos ^ (:r + ?/) sin h {x — y). 

The student shonld be familiar with the following relations 
from his study of the Differential Calcnlus.^ 






x^ x^ 



^o^x^l - ,-V "r rr -i^+ (IS) 



\L 



4 I G 



sm cc = X - rw + r^ - ,-^ + • • • (16) 

o 5 I < 



£'^ 



;:C' 



I 9. 





.r^ 




x' 




X5 




•r'^ 




•r^ 


+ 




■ + 




+ 




+ 




+ 






3 




i± 




5 




6 




1 < 



+ X -h — + -—+ ,— -+ ,-— + ,— -+ ,-^+ . . . (10 



By substituting another variable, i6, {l = \^~l), for x in 
(17) there results 

fQ2 . m m 



£' 



12 13 14 



L^ I ^ 



=(i-^+,-^-...U^/^- " " 



2 j4 / V P |S 

whicli compared with (15) and (16), gives 

£^^ = cos ^+i; sin <9 (18) 

* See Price, Vol. I, Arts. 69 to 61; Chauvenet's Plane and Spherical 
Trigonometry, Chapter XIV; and Peirce's Plane and Spherical Trigo- 
nometry, Chapter VII. 



TRIGONOMETRIC DIFFERENTIALS. 29 

Substituting —16 for x in (17) there results similarly 

£-i^ = cos (9-i sin (9 (19) 

Since e is the base of the ISTaperian system of logarithms, 
(18) and (19) may be written 

/(9 = log (cos i9+i^sin (9) (20) 

and - i^ = log (cos ^ - i! sin ^) (21) 

Adding (18) and (19) gives 

cos6>= ^--^ (22) 

Subtracting (19) from (18) results in 

sin^=^— ^^^ — (23) 



Al 



Dividing (23) by (22) gives 

tan(9=— ^ =— — . . . (24) 

2 Trigonometric Differentials. — Every differential expres- 
sion consisting of the product of integral powers of trigono- 
metric functions of one angle, multiplied by the differential 
of the angle, can be reduced to the form 

d2/ = G0S^ X sin^ x dx, 

in which m and n are integers, even or odd, positive or 
negative, or zero. 

It will now be shown how each of the different cases of 
this expression may be integrated. 



3Q INTEGRATION. 

3 To integrate dy= cos*- x sin- dx when either mornis 
an odd positive integer, no matter what the other may be. 

(a) Let m = 2r + l,r being a positive integer, 
then dj/ = cos^^ x sin^ x dx 

= cos2^+^ X sin^ X dx 

= sin^ X {1- sin2 xy cos a^ ^;^ 

= {sin^a^-r sin^+2^ + . . . ^sin^+^r ^},Z (sin ^), and 

sin-+^ X r sin-+3 x ^m-+^^+^ ^ ^^ 

(b) Similarly, when n is of the form 2r + 1, 
d(j = cos^ cc sin^^ X dx 

= cos^ X sin2''+i ^ (jx 

= cos^ i;c (1 - cos^ xy sin x ^Zx 

= - {cos^ X - r cos^+2 ^ + . . . -i-cos^+2r^j^(cosx),and 

cos^+^ X r cos^+^ X _ -p cos^+^^+ ^ x _^ ^^ 

y ^~ ,;, + ! ^ m + 3 * ' * m + 2r+l 

Example 1. To integrate J^ = sin2 ^ cos^ x dx. 

dij = sin^ X (1 - sin2 x) d (sin x), 
_ sin^ X __ sin^ x ^ 
^" "~3~ 5 

Ex. 2. To integrate c/^ = tan*^ x dx. 

sin^ x dx ^ (1-cos^) g.^ ^ ^^ 
COS^ X cos^ X 

_ (Z (cos x) d (cos x ) 

COS^ X COS X 

7/ = 1- log COS X -f C\ or 

2 cos^x 

= ?^^ - log sec X + C. 



TRIGONOMETRIC DIFFERENTIALS. 31 

EXAMPLES. I. 

1. dy = sin x cos x dx, 10. dy = cos^ x dx, 

2. dy = sin^ x dx. 11. dy = cot^ x dx, 

3. dy = tan x rZa:*. 13. r/^ = tan^ j^ c/a:. 

4. dy = cot a: fZa:*. 13. dy = tan^ a:: sec x dx, 

5. c/y = sin^ X cos^ a? cZcc. 14. dy = tan^ x sec^ x dx, 

6. (f?/ = cot^ a: dx. j^ 7 _ sin^ a: r/a: 

7. (i^ = sin^ cc c?x. cos^ x 

8. ^?/ = cos^ X dx. -.Q J _ cos^ X dx 

^ , sin^ X dx sin^ x 

9. dy = . 

cos X 

4. To integrate djr=cos"» :«rsin" x dx when 222 + 22 is an 
even negative integer. 

(a) Let 771 -\-n= - 2r, then 
dy = cos^ a? sia^ x dx 

= tan^ a:^ cos^"^^ x dx 

= tan^^ x sec-"" x ^Zx 

= tan^ X sec^''~^ x sec^ a:: dx 

= tan^ a: (1 + tan^ xy~^ d (tan a^) 

= \ tan^ a^ + (?' - 1) tan^+2 ^ ^ _ ^ tan^+2r-2 ^ ^ ^/ ^^^^^^ ^^^ 

and, 

tan^+i a: , , ^s tan^^^- ^ tan^+2r-i ^ 

^ ?1 + 1 71+3 7?. + 27'-l 

(6) This integration may also be performed as follows : 
dy = cos^ x sin^ x dx 

= cot'" X sin^+^ X dx 

= cot^ X csc^^ X dx 

= _ cot^ X (1 + cot2 a^)^-i c? (cot x) 

= _ 5 cot^ x-\-{r-l) cot^+2 ^ _|_ ^ _|_ cot^+2r-2 ^.j. ^ (cot a;), 

COt^"^^ X , ^s COt^+^ X C0t^+2r-l ^ 

■^ m + 1 m +3 ?/^ + 2r-l 



32 INTEGRATION. 

Example 1. To integrate cly = — . 

sin X Qo^^x 

■J __ sec^ X dx _ (1 + tan^ x) d (tan x) 
tan X tan x 

= I htana:-) ^(tan :r), and 

\tan X I 

y = logtan x + — - — + C. 

EXAMPLES. II. 

1. dy = sec^ X dx. 6 / _ ^^^ 

2. dy = sec^ x dx. sin- x cos^ x ' 

3. dy = csc^ x dx. ^ . _ sin- x dx 

4. dy = csc*^ X dx, cos^ x- 

e 7 f/x 8. (i// = cot^ X csG^' X dx. 

5. dy = . -^ 

sm X cos X 9. dy = tan^ x sec^ x dx. 

10. % = -^ ^^ . 

sin^ x cos^ X 

5. To integrate djr= cos"' jx- sin" jx- dx when both iii and n 
are even positive integers. When both 7??. and n are even 

positive integers, by doubling the angle as often as necessary 
by means of the following trigonometric relations, 

-' -- ^"'^'' (9, p. 28) 

(10, p. 28) 
(10, p. 28) 

this differential expression may be transformed into an expres- 
sion involving sines and cosines of multiple angles and then 
integrated. 









2 




cos^ 


X = 


1 


4- COS 

2 


2.r 


sin- 


x = 


1 


— COS 


2.7- 



TRIGONOMETRIC DIFFERENTIALS. 33 

This may be best illustrated by means of a few examples. 

Example 1. To integrate dy = sin^ x dx. 

-, 1 — cos 2x 7 
dy = ax, 

X sin 2x ^ 
y=- + C, 

Ex. 2. To integrate dy = sin^ x cos^ x dx, 

7 sin^ 2x 7 

di/ = dx 

^ 4 

1 — cos 4x 7 

= dx, 

8 

X sin 4x ^ 

-^ 8 32 

Ex. 3. To integrate dy = sin- x- cos^ a: dx. 



dy = sin- x cos^ x cos^ x c7x 
sin^ 2x /I + cos 2x 



4 V 2 



r/.T 



sin^ 2x sin^ 2x' cos 2^:^ \ 7 



8 
1 — cos 4x sin^ 2a^ cos 2::c 



dx, 



16 



cc sin 4;:c sin^ 2x ^ 
-^ 16 64 48 



EXAMPLES. III. 

1. dy = cos- x dx. ^. dy = cos^ x dx. 

2, dy = sin^ x dx. 4. dy = sin^ x cos^ x dx. 

5. dy = sin"* x cos^ x dx. 



34 INTEGRATION. 

6. It will now be shown liow the remaining cases of 
d2/ = cos^ X sin^^ x dx may be integrated by one or more of the 
preceding cases, either directly or by means of an imaginary 
trigonometric substitution. 

7. To integrate dy = cos*" x sin** x dx when 222+22 is an 
odd negative integer. 

I. When either vi or n is an even positive integer, 

(a). When n is an even positive integer. 

Let 71 = 2p and m = - {2p + 2r + 1) , 

then m + n= — (2r + 1), an odd negative integer, 

and di/ = Gos^ x sin^ x dx 

= sm^^ X cos~<^^^+-'"'^ ) X dx 

= tan--P X sec^''"^^ x dx 

= tan^^ X sec^''"^ x d (tan x) . . , . (A) 

which last expression may now be transformed into 
^7/= (/)2i^+i sin2^' e cos^^ e dd 

by means of the following imaginary trigonometric substitu- 
tion : 

Let tan x = i sin 6, then d (tan x) = i cos 6 dO, 



and Vl + tan2 x = Vl - sin^ 0, 

or, sec X = cos 6, 

Substituting these values in (A), 

dij= {% sin Qfp cos-'-i B i cos 6 d6 

= (/)2?>+i sin^^ 6 cos-'' 6 dO (B) 

to which last expression the method of Art. 5 is applicable, 
since 2p and 2r are even positive integers. 

As far as the integration is concerned, i is treated the same 
as any other constant. After integration the functions of 6 



TRIGONOMETRIC DIFFERENTIALS. 35 

are replaced by the corresponding functions of x, and iO is 
replaced by log (cos + i sin 6) and - iO by log (cos 0~i sin 6). 
[See p. 29, (20) and (21).] 

Example 1. To integrate chj = sec x dx, 

1 _ sec^ X dx _ d (tan x) 
sec X sec x 

i cos ^ cZ^ 



cos 
and y=iO-\- C 



= i* ^(9, 



= log (cos + i sin 0) + C 
= log (sec X + tan x) + C, 

Ex. 2. To integrate cZ?/ = tan^ x sec^ x f/x. 

dy = tan^ cc sec x d (tan a::) 
= (i sin ^)2 cos ^ i cos ^ cZ^ 
= — ^ sin^ cos^ ^ c/^, 

and y == - i /- - ?i^^ + (7 (See Ex. 2, Art. 5.) 

= -log (cos e-i sin (9)+ !_?lEi_^^(i_2 sin^ 6>) + C 
8 8 

= — log (sec X — tan^) h — - (1 + 2 tan- x) + C. 



(b) When 771 is an even positive integer. 

Let m = 2p and n = - {2p + 2r + 1), then 

dy = cos^ ^ sin^ x dx 

= cos2^ a- sin-(2p+2r+i) ^ ^^ 

= cof'P X csc-''+^ X dx 

= — cot"^ X csc-"*"^ ;r d (cot a:) 

which last expression is transformed into 

dy= - (iyp+^ sin2^ cos^^ 6 cZ^ , . . ( (7) 



36 INTEGRATION. 

by means of the following imaginary trigonometric substitu- 
tion : 

Let cot X = i sin 6, then cl (cot x) = i cos 6 dO, 



and V 1 + cot2 x = \/l - sin^ 9, 

or, CSC X = cos 6. 

(C) is the same expression as (B) with change of sign, and 
can therefore also be integrated by method of Art. 5. 



Ex. 3. To integrate dy = esc x dx, 

■J _ csc^ x dx _ d (cot x) 

CSC X CSC X 

i cos 6 dO . n^ 

cos 

and y= —i 0+ C 

= log (cos ^ — ^ sin 6)-h C 
= log (esc X — cot i:c) + C, 

Ex. 4. To integrate dy = cot^ x esc x dx. 
J _ cot^ X d (cot a:) 

CSC X 

(i sin (9)^ i cos d dd 

COS 

= i sin^ ^ t/^ 
y _ie _ i sin 2 g _^ ^ ^g^^ -^^^^ _^^ ^^^^ ^^ 

It , /^ • • /IN i sin ^ cos ^ ^ 
= - log (cos 6+ I sm ^) + C 

1 T X , X cot X CSC :c ^ 
= — log (esc a: + cot x) h C. 



TRIGONOMETRIC DIFFERENTIALS. 37 

II. When both in and n are negative integers, 
(a) When n is even and 7?i odd. 
Let ?i = — 2/j and vi = — (2r 4- 1), 

then VI + ?i = - (2p + 2r + 1), 
and di/ = cos^ x sin^ x dx 

= cos~(-''+i) X sin~-^' X dx 

-_ dx^ _ sec-P+^^+^ X dx 

sin^^' X cos^''+^ X tan^^ x 

_ sec-^+-''~i' X d (tan x) 
tan^^' X 

Ey use of the imaginary trigonometric substitution given 
in I {a), 

^ {i sin ey^ ^ ^ 

^ (1-sin^^r-^^ 

^ ^ sin2^ 6> 

= (i)i-2^ {csc-^^ 6>- (i9 + r) csc2^-2 e+ , . ± sin^^ 6*} r/(9, 

each term of which can now be integrated by previous cases, 
since all the exponents are even ; the terms involving esc ^ 
by Art. 4, and the terms involving sin by Art. 5. 

dx 



Ex. 5. To integrate dy = 
dy 



sin^ X cos X 
sec^ X dx _ sec x d (tan x) 
tan^ X tan^ cc 

_ i cos'' edO _ _ . (1 - sin^ 0) 

(i sin ^)2 sin^ $ 

= -i (csc2 6-1) dO 
y = i cot 9+i 6+ C 
cos ^ 



^ sm t^ 

sec a? 



+ log (cos 6 + i sin 6)+ C 
+ log (sec a: 4- tan x) + C 



tan 0^ 
= log (sec X + tan x) — esc x + (7. 



38 INTEGRATION. 

(b) AVhen 7^1 is even and ?i odd. 

clf/ = cos^ X sin" X dx 
= cos~^^ X sin~(-^"*~i) X dx 

_ csc^P+^^-+^ X dx _ _ CSC2P+2r-l y, ^ (^^^^ ^^ 

cot^ X cot^^' X 

whicli by means of the imaginary trigonometric substitution 
given in I {h), is transformed into 

. i COS2^+2r Q ^Q 

dt/= • 

^ {i sin eyp 

which last expression is the same as (D) with change of sign, 
and may therefore be integrated in the same way. 

Ex. 6. To integrate dy = 



sin^ X cos^ X 



1 _ csc'^ X dx __ csc^ X d (cot .r) 
COt^ X ' cot^ X 

^ _ i cos^ d6 _ i cos^ i9 dd 

{I sin ^)' sin- 

^./ l-2sin3^^sin^^ X^^ 

V sin 6> / 

= '/(csc2 6>-2 + sin2 (9) r/(9 

= .-(csc2^-2+i-^:^V7^ 

2,= -^ cot ^_3,.^_^sin2^ ^ 
2 4 

cos ^ 3 T . /I • • /IN i sin ^ cos 5 . ^ 
+ -log (cos ^-i sm ^) + C 



i sin (9 2 ' ' 2 

CSC a: , 3 T , , s cot .r esc x ^ 

= h — log (CSC X - cot X) h C? 

cot a: 2 ^^ ^ 2 

3 n . i N cot .r CSC X ^ 

= — log (esc X - cot X) h sec a: + c7. 



TRIGONOMETR-C DIFFERENTIALS. 39 

EXAMPLES. IV. 
1. di)=^ sec^ .r dx, 7. r7^ = cot- x csc^ x dx. 



2. 


c?y = sec^ a: dx. 




8. 


^y = 


COt^ .T CSC X dx. 


3 


dif = csc^ a: dx. 




9. 


^y = 


dx 




sin2 X' cos^ X 


4. 


di/ = CSG^ X dx. 




10. 


^3/ = 


dx 
sin X cos^ X 


5. 


€?y = tan2 ^ sec a: 


(7a:. 


11. 


.7^ = 


dx 
sin^ X cos X 


a 


cfy = tan"* X sec x 


dx. 


12. 


dl/ = 


sin* a: (7a: 



COS^ X 

S. To integrate djr=cos*^ sin" x dx when either m or n 
is an even positive integer, the other being a negative integer, 
and ia + 22 is a positive integer or zero. 

(a) When n is an even positive integer. 

Let n = 2/9 and m = —{2p — r), 

then in + n = r, a positive integer, 

and dy = cos^ x sin^ x dx 

= cos-(2p-^') X sin-2^ X (7^ 

_ (1 — cos- xy dx 

COS^'-^~'" X 

= (sec-^'-*' X -X) sec2p-"'-2 x+ . . . ± cos^ x) (7x . (E) 

(b) Similarly, when m is an even positive integer, 

dy = 0,0^^'^ X sin^ x dx 

_ (1 - sin^ x)^' dx 
sin^p-r ^ 

= (csc-^'-'' X - /9 csc-^-^-2 x + . . . ± sin'' x) dx . (i'') 



40 INTEGRATION. 

(E) and (F) are similar expressions in which the exponents 
of all the terms are even or odd, according as r is even or odd. 
When r is even, the methods of Arts. 4 and 5 are employed, 
and when r is odd, Arts. 3 and 7. 

Example 1. *To integrate di/ = [ L". 

sin X 

dy= ^ — ^^-^ '—^ dx 

sin X 

= (esc cr — sin x) dx 
y = log (esc X - cot X) 4- cos x^ C, (See Ex. 3, Art. 7.) 

Ex. 2. To integrate dy = tan* x dx, 

-J _ sin* X dx _ (1 — 2 cos^ x + cos* cc) -. 
cos* X cos* X 

= (sec* x — 2 sec^ a: + 1) fZa:^ 
= (1 + ta.n2 X) d (tan x) — 2 cZ (tan a:*) + 6Za:5. 

?/ = tan X + — 2 tan x + x + C 

i tian X /^ 
= a: - tan x ^ y C 



EXAMPLES. V. 

1. dy = tan^ x dx, 5. dy = cot^ x dx. 

^ J sin^ X dx r> 7 cos* a: (^Z.x 

2. a^ = . 6. dy = . 

cos X sin^ a; 

o 7 J. A 1 mi sin* .T <f;r 

3. dy = tan* e:c dx, 7. ^^ = 



cos*" 03 



4. <^?/ = cot^ a: rZx. 



TRIGONOMETRIC DIFFERENTIALS. 41 

9. To integrate 

dy= cos mx cos nx dXy dy= sin rax cos nx dXy 
and djr= sin mx sin iix dx. 

dy = cos mx cos nx dx 

= [^ cos (m + ^^) a; + I cos (m - n) x'\ dx (11, p. "2^^^ 

_ sin (m-\-n) x sin (m — n)x ^ 
2 (vi + n) 2 (m — n) 

c?^ = sin mx cos no: cZa: 

= [1 sin (m-\-n) x+h sin (m — 7^) a:] dx (13, p. 28) 

_ cos (m-^7i) X cos (m — n)x ^ 
2 (??^ + ^) 2 (m — n) 

dy = sin ?7^a; sin nx dx 

= [ ~ 2 ^^s (m + n) x-\-^ cos (7?i — 7i) x] c^a; (12, p. 28) 

sin (m-\-n) x , sin (m — n)x ^ 

y= ^^ ^^ + ^^ h 6. 

2 (771 + 7i) 2 (m — n) 



EXAMPLES. VI. 

1. dy = cos Sx sin 5a: c^a:. 

2. dy = sin 5^ sin 6x dx. 

3. dy = cos 4tx cos To: ^Zx. 

4. ^7/ = sin ^x cos fx cZa:. (See 14, p. 2S.) 

5. c?^ = cos fa; sin -Jo; dx. 

6. 6^2/ = cos 3x cos Jo; c^o;. 



42 



INTEGRATION. 



10. To integrate dy= ^^ cos hx dx.^ 

Substituting for cos bx its value in exponential form 
[(22), p. 29], 

di/ = e""' I - — ^ \dx = ±[ e^'«+'^^ ^ + e^«-*^ ^] dx, 






2/== - ^ + 

2\ a-{- ib a— ib 



a 4- ii 

-CUT n ibx 

2 [_a 4- // 

2 L 









+ c 



(g'"^^ + g-'"^-^) _ ^'^^ (£^^-£-^'^^) "[ 



a"- ^h 



{it cos ^:« + Z> sin ^^) + C. [(22) and (23), p. 29] 



This last expression admits of further simplification by 
substitutinor 



cos (0 for 



hence 



, and sin w for — ^ , where w = tan~^ — , 



Va^ + Z>2 



V = 



\/a2 + 52 



(cos o) COS bx + sin w sin bx^ + C 



cos {bx — (o) + (7 , 



Va2 + ^2 

* Price, Vol. II, Art. 76. 



(5, p. 27) 



TRIGONOMETRIC DIFFERENTIALS. 



43 



11. To integrate dy=e*'^ sin bx dx- 

Substituting for sin bx^ its value in exponential form [(23), 
P 29], 



di/ = c«^ [ t _r J Jx = -p-. [s'^+'^"^ - £(a-^^^j dx 



y= -7-J ^^ :: ^^ — I + C 



{ 



2i a + ib a — lb 



2i \_a-i-ib a — ib^ 



2.1 



a£*^ - ibe'^'' - ae"*^ - //>-- 



a^-^P 



+ C 



a? + b' 



2i 



ibx _| ^—ihx 



-f g-^^"j 1 



+ (7 



a- + h 



- (a sin bx-b cos Z^x) + a [(22) and (23), p. 29] 



Va^ + P 



(cos w sin bx — sin w cos bx) + C 



— 1 . sin (bx — a>) + C, where oj = tan~^ - . (6, p. 27) 



44 ' INTEGRATION. 

12. The integration of the differential expressions of the 
two preceding articles may also be performed as follows : 

Let du = s"^ cos bx dx and dv = s""^ sin hx dx, 

multiplying the latter through by i, and adding to the former 

du + idv = £^^ cos bx dx + i e^^ sin bx dx 
= e""^ (cos bx + i sin bx) dx 
= £«^ £*^ rfi^ = £(«+*>^ dx . . . (18, p. 28) 

u + iv = — + C 

a + tb 



a — ib 



e"^ (cos bx 4- i sin 5:c) + C 



a'^ + P 

JCIX 

(a cos bx -\-b sin ^a:*) 

■\ (a sm bx — b cos ^x) + C 

^2 ,7.2 ^ ; -rv^. 



a?-\-b'^ 



Equating the real parts, 



u = — {a cos 603 + 5 sin bx) + C^ 

cos f bx — tan~i — ) + C 



\/a2 + 52 V ^^ 



TRIGONOMETRIC DIFFERENTIALS. 45 

Equating the imaginary parts and dividing through by ^, 

(a sin hx -h cos bx) + C ^ 



a' + h'' 



Va2 + P 



sin lhx-t2in-^-\ + C^. 



EXAMPLES. VII. 

1. dy = £•' cos X dx. 

2. dy= e"" sin x dx. 

3. dy = e-"" cos 3x dx. 

. , sin X dx 

4. dy= . 

5. dy= e~^^ cos 2x dx. 
Find the value of / in the following equations : 



6. // = T^ / e^ sin o) ^^(iz^ + A:. 



7. U = ""^ / ^ cos o) 2^ a?^ + A;. 



46 INTEGRATION. 

MISCELLANEOUS EXAMPLES. I. 

Ix 



1. di/=-siii^ X dx. 17. di/= 

2. df/ = sin^ X dx. 18. dt/ = 



Slll^ X cos'"^ X 



3. di/ = sin^ x cos^ x dx, 19. dij = — 

4. rZ^ = tan a: sec^ x dx, 20. dij = 



sin'' .r 



sin .1' cos^ X 
sin'' a' dx 



Vcos .r 

5. rZy = tan^ X dx, 21. dij = £-^ cos 4x' dx. 

6. cZ?/ = tan^ X dx, 23. ^y = cos 4x cos 5x' tZx. 

7. f/y = cot^ X dx, 23. f/y = x^ sin' (x^) cos^ (x^) cZa:> 

8. dij = cot^ X dx, 24. r/y = tan^ x sec^ x fZx. 

o 7 7 7 or 7 sin^ X c/x 

9. dy = seC'' X dx, 25. c/y = . 

cos X 

10. dij = sec^ X tZx. 26. dy = cot^ x csc^ x dx. 

11. 6Zy = cos^ X 6Zx. 27. cZ?/ = cot^ x csc^ X 6Zx. 
13. (i?/ = cos^ X <:Zx. 28. dy = sin- x cos^ x cZx. 

13. cZy = csc7 X cZx. 29. rZ?/ = ^^^^ ^^ ^^ . 

sill'' X 

14. dy = esc' X dx, 30. dy = ^^^ ' ^^ ^^^ . 

A/sin X 

15. cZy = tan2 x sec^ x dx, SI. dy = cot x esc"' x rfx. 

16. dy = sin^ x cos^ x (ix. 32. rf?/= e^^ cos 4x (^x. 



CHAPTER II. 
RATIONALIZATION BY TRIGONOMETRIC SUBSTITUTION. ^ 

13. Introduction. Since in a right-angled triangle the 
square of the hypothenuse is equal to the sum of the squares 
of the other two sides, \/d'-'^x^ may be represented by the 
hypothenuse, and \^a' — x^ or V^x^ — a? by one of the other 
two sides of a right-angled triangle, and each one of these 
surds may therefore be expressed rationally as a trigono- 
metric function of one of the acute angles of the triangle. 

Differential expressions containing one of these surds may 
therefore be rationalized by transforming into trigonometric 
functions, and, unless otherwise too complicated, integrated 
by methods given in Chapter I. 



14. Rationalization of expressions containing Va^-h^^* 




From the triangle in Fig. 1, 



and 



\^a^ -\-x^ = a sec 6, 
x = a tan 6, 
dx = a sec^ dO, 
* Price, Vol. II, Art. 79. 
47 



48 INTEGRATION. 

dx 



Example 1. To integrate di/ = 

a^ + x^ 

Substituting the values obtained above, 
a sec- 6 d6 _ dO 
a^ sec- Q a 



dy = 



a 
Substituting the values for obtained directly from the 
triangle in Fig. 1, 

y = - tan-i - + C. 



Ex. 2. To integrate dy = 



a a 

dx 



. ^ a sec^ e dO _ de _ c os (9 dO 
a^ sec^ a^ sec 6 a? 

y = — sin 6+ C 
a? 



^2 ^ g2 _p ^2 



EXAMPLES. VIM. 

1. dy= ^f ^^^^ ♦ 7. rfy ^'^^ 



,,2 






(a- + 2-2)5 \/a2 + a'2 

(7x 






5- ^2/=^v^.- 'll. cf;/= ^^•^^■^- 



.2\3 



6. rf2/=-#^,. 12. Jy= -'^- 



(a2+a;2)2 C«2 + a-?)3 



TRIGONOMETRIC SUBSTITUTION. 



49 



15. Rationalization of expressions containing Va^ - ^^, 




rrom. the triangle in Fig. 2, 



and 



\/ ci^ — x^^a cos ^, 
x = a sin ^, 
dx = a cos dO . 



Example 1. To integrate dy = -j^ 
a cos 6 dO 



dx 



dij = - 



a cos 



de 



3/ = 6>+(7=sin-i- + C. 
a 

Ex. 2. To integrate dij = 



x^ dx 



\/a2- 



^^ ^ ^3 sin3 ^ g cos dO _ ^^3 si,,3 ^ ^^ 
a cos 
= a^ (1- cos^ ^) sin dO, 
a^ cos^ ^ 



y=—a^ COS c^ 
a^ COS ^ 



~+c 



(3 - cos2 ^) -h (7 



iWa'- 



+ c 



Va^ — cc^ 



(2a2 + a;2)+C. 



60 



INTEGRATION. 



1. dy = 



EXAMPLES. IX. 

x" dx ^ ^j ^ (l-xy^dx 



Vcr--x-' 



2. di/= \/«'-*-.i-2 dx. 

3. d>j= -^'^-^ 



4. chj= {(i'--x")i dx. 
.T^ dx 



8. du 

9. dy = 
10. ,/// = 



dx 



x^ Va- — x^ 



{(i^-j?f^ 



5. dy = 

6. rf?/ = 



x^ dx 

(a^-xy 



11 . dy = x^ x/ct^ - x' rf.r. 

12. du=x^{l-x'Y^ dx. 
dx 



7. rf^ = 



\/a2 — x^ dx 



13. rf^ = 
14 6/y = 



X (a^ — X') 

x^ dx 

{C(?-X')i' 



10. Rationalization of expressions containing \/^^ - a^ 




Fig. 3. 
From the triangle in Fig. 3, 



and 



\/x^ — a? = a tan 0, 
x = o. sec ^, 
dx = a sec B tan ^ rf^. 
dx 



Example 1. To integrate dy = 
dtj = 



x\^x^ — a^ 
a sec 6 tan ^ dd _ r/^ 
a sec a tan ^ a 

y= - +C= -sec-i^ + C. 
a a a 



TRIGONOMETKIC SUBSTITUTION. 51 



/ •> 2 

Ex. 2. To integrate dy = ^-^^— -^c/.r. 

X 

7 a tan ^ a sec ^ tan ^ c?^ 



a sec B 
a tan2 dO=a (sec^ i9 - 1) (i(9, 



?/ = (X tan — aO + C 



= Vx^ - a'^ - a sec-i - + C. 



a 



XoTE. The other angle of the triangle could also have been 
used, in Avhich case, calling this angle a, 



\^«^ + X- = a CSC a, x= a cot a, and dx = — a esc- a da. 
\^ cC- — x^ = a sin a, x = a cos a, and c?a: = — a sin a da. 



V'x^ — a^ = a cot a, x = a esc a, and dx = —a cot a esc c; da. 



EXAMPLES. X. 



^ 7 \/a:^ — <^^ dx jr 1 \/:r — a^ dx 
1- ^y= -^ ^' rf.y= -. 

XT X' 






3. dy= ^ 7. rfy= ^'^ 



X^\^X^ — O? X'^\'^ XT — O? 



4. dy^ (x'-ar^dx^ 8_ ^^^ ^^-^'-f^^^ 



62 INTEGRATION. 

1 7. Change of form of radical. In some cases rationalizing 
a differential expression by trigonometric substitution as just 
shown, leads to a trigonometric differential expression which 
would be integrated by means of an imaginary trigonometric 
substitution, which, of course, can be done. 

On the other hand, since the form of the radical determines 
the resulting trigonometric function, it is possible, by using 
the imaginary, to change the form of the radical so that the 
resulting trigonometric differential expression may at once be 
integrated. 

The sign of both terms under the radical may be changed 
together by factoring - 1 from under the radical, thus : 



\/a^ + x^ = \/ — 1 { — a'^ — x'^) = i\/ — a^ — x^y 



Vx2 -a^ = \/ -1 {a^-x'^) = i \/a^ - x\ 

Also by putting x = iv, the sign of x^ can be changed alone 
thus : 



\/x'^ - ^2 = V - 



■a\ 



Hence it is possible to change the form of the given radical 
into any one of the three forms, as follows : 



Va^ 






-a? 


■■ i \/y2 


-a2, 


Va? 


= Va2 


+ v\ 



the form chosen being the one which 'renders the simplest ex- 
pression to integrate. 



TRIGONOMETRIC SUBSTITUTION. 53 

It will now be shown how this applies to a number of 

examples. 

dx 
Example 1. To integrate dy = , . 

This expression, if rationalized as shown in Art. 14, would 

reduce to 

dy = a sec d dd 

and integrated same as Ex. 1, Art. 7, but by changing the form 
of the radical to x^a^ — v"" by putting x = iv and dx = i dv, a 
much simpler expression to integrate results, thus : 

dx i dv 

dy = 



^/a} + x^ Vct^ - v'^ 

= i±^21±i^=ide, (See Fig. 4.) 
a cos 6 

y = i e + C = log (cos + i sin 0)+ C 

f\/a^ — v^ -V iv\ 






c 



-\- X -\- X \ . r^ 




a 

i dv . . V 

^dy= . may also be integrated as y = i sin~^ — , 

V ^2 -v^ a 

then 1 = sin 1^ = '^-^^ (23, p. 29.) 

a i 2i 

a a 

multiplying through by e^, 



a 



* Price, Vol. II, Art. ^Q. 



5-i 



INTEGKAIION. 



from winch 
and 



X + V (l^ + !•- 



7/ = l0c 



.r + 



\^d^ + : 



Ex. 2. To integrate (/^ = 



(7- 

dx 
o? — x^ 



+ C. 



Let 
and 



X = i??, then dx = l dv, 



chj- 



i dv 



i a sec'- edO i dO 



h ^2 a^ sec^ 6 a 

y^^ll^C =-log (cos e^i sm e) + (7 



(See Fig. 5.) 




= — loe 



- log 



a 4- I'V 



ci^x 
\/d^ — 1 



+ c 



+ c 





a 




Fig. 5. 




dy- :^^c 




^2 -h ?;2 


len 


- = tan -^ : 




(2 ^ 


iv 


_ X _ £2«'^ - 1 



= i log . Z^+i:' + C. 
« V a - X' 



can also be integrated as ?/ = — tan~i —y 

a a 






By composition and division in proportion, 



(24, p. 29. > 



9^2az/ 



g + a: 
a — a; 



2a?/ = log 



a — X 



and 



2/= - log V/ + C'- 

a V a - a: 



TRIGONOMETRIC SUBSTITUTION. 



55 



Ex. 3. To integrate dy = 



Va?- 



■dx* 



dy = 



i \/:i 



I \/ X — d 



dx 



i a tan 6 a sec 6 tan dO 



(See Fig. 3.) 



a sec 6 

= ia tan^ dO=ia (sec^ ^-1) dO, 
y = ia tan Q — iaO + C 
= la tan 6-\- a log (cos 6 -i sin 6) + C 



= ^ Vx^ — a^ + a log 



a — iwx^ — ( 



■c 



= Va2 - X2 + a log / ^--V^^-^ ^\ ^ ^^ 



Ex. 4. To integrate d?/ = 



(ia: 



Vx2-. 



dy = 



dx 



i \/ a^ — x^ 

a sin dO 
ia sin ^ 



de 



(SeeEig.6.) 



3/= - ^ +c = ie+c 

= log (cos 6^1 sin ^) + C 
_ log f x+iVa^-x '\ ^ Q 

=iog^^±^/?!^^'Uc. 




Fig. 6. 



^/a--. 



66 INTEGRATION. 

dx 
Ex. 5. To integrate dy = =^ 

x Vct^ + x^ 

Let X = ivy then dx = i dv, 

and dy = 




t dv 



dv 



iv\/a'^_y2 io\/v''--a? 
a CSC 6 cot ^ rf^ 



ia CSC ^ a cot 

d0 _ i dO 
la a 



(See Fig. 7.) 



rig. 7. 



ii + C ^ 1 log (COS ^ 4- i sin 6) +C 



-log 



V'^;^- 



a'' I a 



V 



+ ^]+C 



= _ log 



-C 



a \ X 



Ex. 6. To integrate dy 

Let 

and dy = 



dx 



(x^-<x2)^ 
X = iv, then dx = i dv, 

i dv i dv 



dv 



a sec2 6> dO cos 6 d^ 



2/= - 



a^ sec^ 6 
sin Op 



, (See Fig 5.) 



aVa2 + v2 



+ (7 



aH\/a? — x^ 



+ C 



a?\/^ — 6^2 



+ a 



TRIGONOMETRIC SUBSTITUTION. 67 

EXAMPLES. XI. 

11. dy= ^^ 



1. 


ay = 


\/a^ 4- x^ dx. 


*>, 


dy = 


dx 




{a'-xy 


3. 


dy = 


x^ dx 




Vx^-a^ 


1, 


dy = 


dx 




x(a^-x^)i 


5. 


dy = 


dx 




x\a^-xy 


6. 


dy = 


(1 + xy-dx, 



dx 



13. dy = 



x\/a^ - x^ 
x^ dx 



{x^ - a?) 



13. dy = V"^' + ^' dx ^ 

x^ 

14. dy = x^ Vl + x^ rfx. 



15. dy = 



16. rf?/ = 



x^\^a^ — x^ 

x^ dx 
(a'-xy' 



r^ n \/l + x^dx ^^ , \/a^-x^dx 



x"" 
dx 



8. dy = xWx^ - ^2 dx. 18. dv= —^ 



9. dy= -^^ . 19. dy = Vx^ - a^ dx, 

x^ — a^ 

10. dy = — ^2 30. dv = -^' ^^ 



58 INTEGRATION. 

MISCELLANEOUS EXAMPLES. IL 

^ , x^ dx 





u.t/ — 


{"' 


+ x')' 


*>, 


dy = 


(.r^ 


- «^)» dx 






X 


3. 


dy = 




dx 




.T(. 


r-«2)i 



x-{a-- 


■x^)i 


dx 




x*{a-- 


x^-) 


dx 




x-{'i- + x^) 


dx 





13. dy = x^Va^ + x^ dx. 



13. dy=x'ix--a'-)idx. 



14. dy^ ^-^ . 

■ x\a--x-)i 



4. <v = x'Va^ - X- dx. 15. dy = (Q''--'g')^c?^ . 



5- rf^/ = .... .!^'' ..3 • 16- ^2/ = '^■'^ 



6. (iv = — . 17. rfw = 

x\a--x^) 



8. dy = — ^i^^; 19. dy = 



9. rf(y= (.r2-a2)irfx. 20. A/= 



^ -I- T- -I. «2 



(«^- 


-x^)^- 




dx 


a;4^/«^-a;2 


.t' 


dx 


(«2- 


-x-'y 


x' 


dx 


(«^- 


-xy 


.T^ 


dx 



10. dy= x*Vx'-a'-dx. 31. (jy = ^ + x- + a ^^ 

{a- + x^)^ 



11. dy = /''^ ,, „ . 33. dy = ^^ 



(.c--«')" x(a+ix^)' 



23. dy = 



TRIGONOMETRIC SUBSTITUTION. 
dx 



59 



Let x = a sin d, then dx = a cos <9 d/9 and 

7 d^ sec2 d 

dy^ 

When a > 6, 

dy- 



a cos /9 V6^ - a^ sin^ ^ aVt^ sec^ d-a^ tan^ ^ 

d (tan 6) 
a Vb^ - (a^ - b^) tan^ d 



2/ = 



a Vo? - ¥ 



aVa^-b^ 



^_i / Va^ - 6^ tan d \ ^ ^ 
\b\/a'-xy 



When a < 6, 



d?/ = 



6? (tan d) 



a \/¥ + (62 - a^) tan^ ^ 



y = ^ ]oJ ^b^ + (b^ - a^) tan2 ^ + V^^ - a" tan g \ ^ ^^ 

aVb'-d' \ b / 

a\/62_a2 ^ [ 



bVa^ 



18. Rationalization of expression containing \/2ax~x'^. 

This surd may be rationalized in two ways, either as 

\/x \/2a — x or as \/a^ — (x — ay. 

In the first case, from Fig. 8, 



\/2a — x = \/2a cos 9, 
x-=2a sin^ 0^ 



\/2ax — x^-=2a sin 9 cos ^, 
and dx = ^a sin ^ cos ^ dO. 




60 



INTEGRATION. 



In the second case, from Fig. 9, 




\/a' 



ix—a) 



(x — (x)2= a cos 6, 
X — a = a sin 6, 

x = a (1 + sin 6), 
dx = a cos 6 dO. 



V2ax^x' 
Fig. 9. 

The latter case is the simpler one, and can always be used 
except when the radical appears in the denominator, multi- 
plied by some power of x, when the first case is used. 

dx 
Example 1. To integrate dy =^ / '=' • 

\/2ax — x'^ 

J 4.a sin 6 cos dO 
dy = 



2a sin 6 cos 
y=2e+C 



= 2de, (See Fig. 8.) 



= 2 sin-i\/^ 
\ 2a 



+ C, 



or 



, a cos 9 dO 7/1 

dy = — = de, 

a cos 

y=6 + C 
= sin-i/^ - iVc. 



(See Fig. 9.) 



19. Rationalization of expressions containing \/2a^+ x^. 

This surd may be rationalized, either as \/x \/2a + x or 

From Fig. 10, 



\/2a + a: = \/2a sec ^, 
x = 2a tan^ ^, 



\/2ax-\-x'^ = 2a sec B tan 6, 
and dx = 4a tan 9 sec^ 9 d9. 




TRIGONOMETRIC SUBSTITUTION. 
From Fig. 11, 



61 



\/2ax + x'^ = a tan 6^ 
x-[- a = a sec 0, 

x = a (sec ^-1), 
and dx = a sec tan dO. 



Example 1. To integrate dj/ = 




V2ajci-x'^ 



dx 



or, 



{2ax + X-) i 



d'y = 



4:a tan sec^ dO 
8a3 sec3 tan^ 

dO 
2^2 sec 6 tan2 

cos^ ^ c/0 1 /I - sin^ j 



(See Fig. 10.) 



2a'^ sin2 2a''-' \ sin^ 



y=^^(-csc ^-sin (9) + C 



d (sin Q)^ 



-TMn^-^^y^ 



X 

2(a + x) 



+ C = - 



a + .T 



c??/ = 



2(x2 \/2ax + x^ 

« sec ^ tan dO 
a^ tan^ 

1 /_cos_^\ ^^ _ Lcot (9 CSC ^ di9, 



+ (^; 



a^\/2ax + x^ 

(See Fig. 11.) 



y= CSC ^ + C 



^ \/2ax + ^2 



+ c 



2 
1. 


INTEGRATION. 
EXAMPLES. XII. 

du ^"^-^ . 6. du- ^^ 




\/2^? X - x^ X ■V2ax - x^ 


2. 


7 X ax ,^ • /^^ o /I ^^• 




^'-^«-^' ^ '(See Fig. 9.) 


3. 


All- ^^^ ■ 7. dv- V>«r 1 t2 r/.r 




\/2ax-x'' 




= i\/a2-(a; + «)2 rf:c. 


4. 


(i^ = \/2ax - x'^ dx. . Let (x + a) = a cos (9. 



5. dy = x \/2ax-x'^dx, 8. (i^ = 



X dx 



v; 



ax — x^ 



20. Rationalization of expressions containing trinomial 
surds. 

Differential expressions, not too complicated, containing 
trinomial snrds may be rationalized and integrated on a plan 
similar to the one followed in the two preceding articles. 

dx 



Example 1. . To integrate di/ = 



{x' + 2x + 10)l 
dx 




(x+l) 



[(a:+l)^+ 9]i 
3 sec^ 6 de 
27 sec^ 

cos dO 



(See Fig. 12.) 



y = 



9 

sin 



+ c 

x+1 



9\/x2 + 2.r + 10 



+ C. 



TRIGONOMETRIC SUBSTITUTION. 63 

21. Binomial Differentials. Every binomial differential 
expression may be reduced to the form 

Z 
X"* {a + hx'^yi dx, 

'where 771, n^ p, q are integers and n is positive. 

As found in text-books this binomial differential expression 
may be rationalized and integrated in the following cases : 

Case I. When = an integer or zero, by assuming 

n 

a + hx'^ = ^^. 

Case XL When + ^ = an integer or zero, by assuming 

n q 

a + hx^ = z'^x'^K 

The student should now be able to prove that this binomial 
differential expression can be rationalized by trigonometric 
substitution in the cases just given, and therefore it is not 
necessary for him to remember these conditions, since he can 
readily determine by trial whether a given binomial differential 
expression can be rationalized by trigonometric substitution. 

dx 
Example 1. To integrate dy = 



0:2 (,^2+ ^^5 



Let \/a^ + x^ = a sec d^ 

then X = a^ tant 0, 

and dx = % a^ tan~3 Q sec^ 6 d6, (Draw the triangle.) 

2dO 



dy = 



3a^ tanS sees 6 
2 cos3 dO 
3a^ sin5 6' 



64 INTEGRATION. 



- sm-3 — 4- C 









2a^+3x' ^^ 



2a^ x {a? + x^)-^ 



EXAMPLES. XIII. 



1. dy= — — . 5. dy = x^ {l-^-x"^)^ dx, 

\/2 + 3x' + a:^ 

2. dy^ ^^ , 6. rf^= {^x-^2)dx 



3. dy = V2 + x~ x'^ dx. 7. dy = 

^- ^^= 7^ — ^ - ^- «V/ = ^^'' («'--')' ^^• 

{ct^^-x^y^. 



ANSWERS. 65 

ANSWERS TO EXAMPLES. 

EXAMPLES. I. PAGE 31. 

^ sin2 X , ^ cos- .T , ^ 

1. 7/= ^-— +C, or 7/= - — ^^ +C. 

o COS^ X r-t 

^' y= TT ^^^ -^ + ^' 

O 

3. ?/ = — log cos X + C, or y = log sec x + C 

4. ?/ = log sin x + C, or ?/ = — log esc x -f C 

- sin^ X 2 . - sin^ x ^ 

5. ?/ = sill'" x+ vC. 

3 5 7 

6. 2/ = logcscx- ^^^+C. 

7 cos X O r rj ,,^ 

• y = — — cos^ X ^ cos^ X — cos X -f C 

7 5 

c . Sin^ X r^ ck COS- X 1 x^ 

8. 3/ = siiix hC. 9. y= logcosx + C. 

o Z 

10. y = sm .r — sm^ x-i sm^ x — h C 

5 7 

. 7/ = log CSC X CSC^ X H CSC^ X — h C 

12. ?/ = ^ — sec^ X + log sec x + C 

4 

^^ sec*^ x , ri ^A sec'^ X sec^ a^ ^ 
13. y= sec x + C. 14. ?/= — ^ hC 

15. 7/= — -— +2 log cos X 7^—-^C. 

16. y=~ CSC X — sin x + C 



66 INTEGRATION. 



EXAMPLES. II. PAGE 32. 



■j-n Y1 3 /y« 

1' y = tan x-\ — + C 

o 

2. y = tan x H — tan^ x + + C 

3 5 

3. ^ = - cot X - 55^ + C. 

o 

4. y = - cot X - — cot^ a; h (7. 

3 5 

5. y = log tan x + C 

6. 2/ = - cot X + tan :r + C 

7. ^ == tan3x ^ ^^ 

o 

e cot^ X 2 ,. cot^ .T ^ 

8. 2/ = -.^ -^ cot^ X - + (7o 

O O i 

10. y=- cot a;+ 2 tan a;+ ^— + C 



EXAMPLES. III. PAGE 33. 

1. ?/ = | + ^/-^ + c. 

2. 3:^ sin 2a; , sin 4a; ^ 

o 3x sin 2j; sin 4x ^ 

^ 8 4 32 

. X sin 4:X sin-"^ 2x' ^ 

^ 16 64 48 

- 3r sin Av sin 8r ^ 

^ 128 128 1024 



ANSWERS. 67 



EXAMPLES. IV. PAGE 39. 

1 1 , , , tan X sec X , ^ 

1. y = ~ log (sec x + taii x) H + C 

3 5 

2. y = — log (sec x + tan x) H — tan x sec x 

8 8 

' tan^ X sec x ^ 

--» It/ i s. COl X CSC X , /-v 

3. 1/ = - log (esc x-cot x) - 4- C 

2 2i 

4. y = — lo.^ (CSC X — cot x) cot X CSC X 

^ S 8 

COt^ X CSC x ^ 

- 1 1 / 4. x , tan x sec x , ^ 

5. y = — log (sec x - tan x) -\ h C 

6. y = — log (sec x + tan x) tan x sec x 

tan^ X sec x ^ 

8 8 4 

8, ^ = — log (esc X — cot x) + — cot X CSC X 

^8 8 

COt^ X CSC X ^ 

-■ s +^- 

*-v o 1 / .1 \ ban X sec x ^^ 

9. V = t; log (sec X + tan x) + esc x + C 

10. y = sec X + log (csc x - cot x) + C 

11. y = log (sec x + tan x) -csc x- ^ 4- C 

o 

io ., 1 i^r. /o^^ , 4-0.. \ ^^^ ^ sec X tan^xsecx 
^^* 2/= 7^ log (sec x + tan x) — 

tan^ X sec^ x ^ 



68 INTEGRATION. 

EXAMPLES. V. PAGE 40- 

1. ?/ = tan x-x + C. 

2. y = log (sec x + tan x) — sin x 4- C. 

3. y = — ~ tan X + X + C, 

o 

4. y = - cot x — X + C. 

. cot^ a' cot^ X . ^ 

5. ?/ = ^ — + — ^ cot X - X + C 

o 3 

^ 3 T , , . cot X CSC X . XV 

6. ^ = - log (csc X + cot x) ~ cos X + C. 

^ tan X sec x 3 t , ^ x • ^^ 

7. ?/ = + - log (sec X - tan x) + sm x + C 



EXAMPLES. VI. PAGE4L 

. cos 8x cos 2x ^ . cos 2x 3 2 ^ 

^' y — 16 r" + ^- -^ 4— +j'^«^3--^+c- 

^ sin 11 X , sin x , ^ ^ cos x , 1 . ^ 

2/= 2^+ '^""^^•. ^^ 2~ ^os-x+C. 

o sin llx sin3x ^ ^ 3 . 13 3.5 ^ 



EXAMPLES VII. PAGE 45. 



1. V = -^ f cos X - 4 ) + C'- 
^ V2 ^ ^ ' 



TT 



2. w= -^ sill .r --+<?. 

c^a: / 3\ 

3. y = -^^zz. ( COS 3x - tan-i _ + C. 

V13V 2/ 



ANSWERS. 69 



4. y=- _i_ sin (x + -\-^C, 



.-3X / 3 



5. 7/=- -^-= sill f2x-tan-i-]+C. 



VT3 V 2- 

6. /= ^"' sin Lz'-tan-^— Vz;c"^^ . 



7. /= — ^"' sinL^^ + tan-i — \^ ke"^ 

i 1 V -^Co 



MISCELLANEOUS EXAMPLES. I. PAGE 46. 

1. 2/ = - COS a; + - cos^ x - ■ — ^ h C 

^ 5x sin 2a; sin^ 2x 3 sin 4.r , ^ 

^ COS^ .T COS'^ X ^ 

S. 7/ = —^ -— 4- C 



sec^ X , ^ 
4. y=__+C. 

o 3 

^ sec^ .T 3 sec^ x 3 sec" x t ^ 

6. y = -— . — + — log sec X + C. 

6 4 2 

7, y = ^ + cot re + .T + C 



8. y = + CSC- a; — log esc .t 4- C- 



7 INTEGRATION. 

5 11 

9. y = — log (sec x + tan x) + — -: tan x sec x 
^16 16 

3 . , , tan^ X sec3 x , ^ 

+ — tan^ X sec x H h C. 

8 o 

10. y = tan a: + tan^ x -{- - tan^ a: + — 1- C. 

^ 5 7 

^^ . 2 . „ sin^ X ^ 

11. ?/ = sm a;- - sm^ ^ + — h C 

3 5 

^o 5a: , sin 2x sin^ 2x , 3 si q 4a; , ^ 

12. y = \- + C . 

^ 16 4 48 64 

13. y = -^log (esc a; — cot x) cot x esc x 

^ 16 ^ ^ ^16 

3 ,o cot^ .T CSC^ X . ^ 

cot^ a; esc x + C- 

8 6 

14. y = - cot a: - cot^ a; - — cot^ x — + C 

5 7 

^ - 1 T . , . tan X sec x tan^ x sec ar 

^^- ^ = T^ ^^^ (^^^ a;-tan x) + — ■ + 

Id lb 8 



tan 3 X sec^ x ^ 



6 

^^ o.r sin^ 2x sin 4x sin 8x ^ 

16. ly = — + C 

^ 128 48 128 1024 

17. ?/ = tan x-2 cot x h C 

o 

^c cot^ X COt^ X , ^ 

18. ?/ = h C . 



19. 



2/ = ^ + sec x + log (esc X - cot x) + C. 



3 



/ L 2 cos2 X , cos^ x\ , ri 

20. 2/ = - 2 Vcos X (1 - — ^ — + ~Y~) 



ANSWERS. 71 



21. y= -^-^cos (4x-tan-' 2) + C. 

^ V 5 

„^ sin 9x , sin x , ^ 

22. y= -^+^^ + C. 

^^ sin^ o:^ sin'3 x^ , ^ 

23.2,= -^ ir- + ^- 

^ - sec^ X 2 sec^ x sec^ a: . ri 

■25. y = log (sec x + tan x) - sin a: h C. 

o 

„« In, , . cot a: CSC x , cot^ .t csc x 

^' y= jq ^^^ (^^^ ^ - ^^^ ''^) + — 1^ — + — s — 

cot^ X CSC^ X p 

6 +^- 

27. 2/ = — log (CSC :c + cot x) - <i5!A^!^ _ ^^^^^ ^ '^^^ ^ 
^ 16 ® ^ ^ 16 8 

6 ^^* 

^Q 5x , sin^ 2x sin 4x sin "ix , ^ 

28. 'z/ = — — + (7. 

^ 128 48 128 1024 

■29. y = —log (CSC X - cot x) + -cot x csc x - ^^^' ^' ^'^^ ^ 
8 8 4 

+ cos :r + C 

/ /i 2 . , sin^ x\ ^ 

•30. y = 2 Vsinx fl - ;^sin2 x + "q— )+ ^' 

31. ^ = - 5^ + C. 

■32. y = ^ COS /4a:-tan-iiV C. 



72 INTEGRATION. 



EXAMPLES. VIM. PAGE 48. 
1. ?/ = — h C . 

3. ^= -log -—=^=-+(7. 
a va^ + x^ 



4. ?/ = log , + C. 

- 3(x- a; + 2 x^ , ^ 

^ 3a4(a2 + x2^)| 






7. y=a* log v'a' + x^ ^ (a^' + x^) (.x^-Sa^) _^^^ 
a 4 



8- y= ^^' + ^' (a;^-2a^)+C. 



9. y= Il+£!H(3x2-2)+a 
15 

10. y= — tan-i ^^^ ^ + C 

^ ^a a S {a' + x') 4 {ct' + x^Y 

4 (^2 + 1)2 



ANSWERS. 73 

EXAMPLES. IX. PAGE 50. 



-, a^ ■ _, X X \/a^ — x^ , ^ 
1. y= - sm 1 +C. 



2. y=«%m-'£+^^^'-^%C. 



3. y = - ^f ^' (8a^ + 4ft^ a;' + 3x*) + C. 

15 

4. t,'= ?|^ sin-i 2. +£v'a^_x2 (5a^-2x^) + C. 

o do 



5. y= ^ -sin-i^ +(7. 



6. ^=.77^ +log^^^^^l^+C. 
2 (a^ — x ) <:6 



X a 



^ (1 - x2)t , (1 - x')^ Vl - x' . _, ^ 

ox^ ox** X 



9. ^ = - ^^^^+C. 10. y^ ^__+C. 



11. y= ^ sin-i^, + I Va2-x2 (2x2-^2) +(7. 



13. i/= 1-sin-i x+:^\/l-x2 (2x2- 1)+ ^'(l-x2)i + C. 
lb 16 b 

13. 2/= i-^ log — i=- +c. 14. 2/ = ..^f;^^' +C. 



74 INTEGRATION. 

EXAMPLES. X. PAGE 51. 



!• V = -7r~ sec ^ - h C 



3. 2/ = — sec-^ ^ + '^'~^' (2^2 - 5:^)2) ^ g^. 
8 a a 8:r^ 

3. ^ = ^ + C. 

4. 7/ = A'-^^' \ ^0:2 - a'-\-a^ sec"! - + C 

\ 3 / a 



5. ly = sec ^ f- -^^ ^ + ^ ^ + C . 



6. ^= ^^_^(:c2^2a2)+C. 



'• ^=^^^^^^'+"')-'^- 



^- ^ = ^3--^!-^^!^(-^--^-^)+^- 



8a^ a 8(x2 X' 



EXAMPLES. XI. PAGE 57. 



1. y = I' log ( ^"^+/+ ^ ) + I ^^-^^77^ + c. 



8. w = — log i/'i±^ + ^ + C. 



3. y = I log (^±XJ^y xVx^-a^ ^ ^_ 



ANSWERS. 75 



a' V a- -x^ a^ \ x J 

fir 3 1 /<:^ + x , X 1 , ^ 

5. y = — log \/ + h C 

^ 2^5 V a - a; 2^^ (a'' - x^) a' x 

6. 2/ = I log (x +\/T+~?) + ^^\^''' (5+2a:2) +C. 

8 o 



,.y.l,^l^i±^y^ii±jL,c. 



2 " V a; / 2x2 



8- 2/ = -g log ( ^^ j + g (^^ - ^ ) + C'. 

9. ^= 1 log i/^:::!' +C. 

a \ x + a 

10. 2/ == A log i/«±J + ^:? + ^ +C. 

11. 2/ = - log + C 

13. w = — log v/^^ 2 +C. 

13. 2, = 1 log / ^+-^^ _^g±Z (.,. + 2a^)+C. 



Vi 



14. 2/ = ± log (\/l + a;2- x) + Z, (1 + 2x2) +C. 



15. 2/ = A log (a-Va^-x^\ Va^-T^ +(.. 
^ 2«3 ^ V X / 2a2 ^2 

16. 2/ = J- log t/i^ + /I (a^ + x') + C. 



76 INTEGRATION. 



18. y = J- log J^ - ^ + C. 



19. y= ^ log/ ^- ^x'-^' ]+ xVx^-a^ ^^^ 
2 \ a J 2 



30. y = i^ log /Vcc^ + x'-x\ ^ xvV + xf ^^^ 



MISCELLANEOUS EXAMPLES. IL PAGE 58. 



1. t^ = 4 tan-^ ^ ^-^ ^^' 



16a a 16 (6^2 + a;2) 8 {a?^rxy' 

o? x^ 

6 (6^2 + ^2^3 



+ . ^'^\ +C. 



^- 2/ = -^^ — p — ^^ — + o.^ Vx^ -a^-a^ sec-^ - + C 

5 3 a 

3. y = - , .L_ - isec->^ +(7. 

16 a 16 8 

6 '■^• 

5. y = + C. 



ANSWERS. 77 

6 _lij,./« + a; 1 1 

7- 2/ =- -f- --3 tan-i ^+C. 

8. 2/ = A tan-i ^ + ^-^ 1. +c. 



9. y^^4 log (-^^^'--^ _ ^J^^^^. 
8 \ a / 8 

4 



10. 2/^ -log^ j + -^-i6 + ^ (^='-«^) 



b 



11. 2/ = A log V A-^ + 5^ ^ +C, 



-^ 16 ^\, a / 16 



a^ 0? 



8 o 



13. y= ^ log /--^ + ^/^^ - «^\ _ £ljL V^^^^2 
^ 16 ®V « / 16 



8 o 



a^W a^ — X? a^ X 3a^ x^ 

ooa^ X^ 

tore. 



78 INTEGRATION. 



16 v= J- lo<^ i/'-^+-^' + '^ + ^^ 

3.3 



17- y= -^^^^(«'+2x^) + c- 

18. 2/ = 2a- loglii-:L + f^ -^ + C. 

19. y = h 

^ a / ^f. .,2\S /I /^2 .,2\2 



6 ((^2 _ ^,2)3 4 (^2 _ ^2^^2 2 (6^2 _ .^2) 



+ log vv^ +e. 



80. 2/ = ^' log / x+ V«^ + xV xVa-^ + x- (2x-^-3a-) + C. 
o \^ CI J 8 

21. 2/ = ^^ +log:^^:^!±i + itan-i^ + C. 
a^ + X a, a a 

33. „ = ^^! + J_ log ^^' + c. 



EXAMPLES. XII. PAGE 62. 



1- 2/= -2-sin '(- - Ij- • {x + 3a)+C. 

2. y=— sin-i (^ - l\ - 4aV2ax-x2 _ §. a (x - «) \/2«a;-x' 



+ 3 +C. 



3. 7j= asin-^(- -l\-V2ax-x^+C. 



ANSWERS. 79 



^ a? . _Jx ^ \ (x - a) \/2ax — x^,^ 

^ 2 [a ) 2 3 



y = — , + c'. 

a\/2ax — x^ 



^ _ ^^^ 1 / .r ^a- \/2 ax + x^N (0:4- a)\/2ax + ^'^ , ^ 
2/ = ^ siu-i /-- - 1 ) - Vax -x^ + C. 



8. w = 



EXAMPLES. XIII. PAGE 64. 



1. y=log{2x + 3 + 2V'2 + 3x + x^) + C. ^ 

4. w = ?^ + C. 

6. y= (2x-3)(3^+2) _26^^^^_,2^ ^^_ ■ 

3 (x^-3x + 3) 3\/3 \/3 

7. 2/= - ^-^ + -i_ log -i:^ + C. 

wa2(a + bx^) no' a + bx"- 






iSC" ^}$^ °J!S^ v.^f'C" ^.5^^ v.^i's" v^t^ V-S^ v!5t^ ^te'^v" *vwC" v^'C^ ^j$i\^ ^)$^ ^t^i^ 



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jllatJ)ematits for 
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Camegte Cccj)ntcal ^cf)oolg 

By PROF. S. S. KELLER 

{Carnegie Technical Schools) 



THE Carnegie Technical Schools Series of 
Mathematical Textbooks has been writ- 
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Algebra 
anti Crigonomettp 

Carnegie Cec{)ntcal ^ctioolg 



g^ertesi 


8vo. Half Leather. 


250 Pages. Price, 1^1.50 Net. 


Algebra Section. M3 Pages. 


Special Separate Edition, ^i.oo Net. 



CONTENTS 

ALGEBRA : Algebra and Arithmetic — Factoring 
— Equations — Graphics — ^The Binomial Theorem — Surds 
— Indeterminate — Equations — Quadratic — Equations — 
Logarithms — Inequalities — Progressions — Interest and 
Annuities. TRIGONOMETRY: The Relation be- 
tween Angle and Line — Solution of Triangles — Solution 
of Oblique Triangles — Spherical Trigonometry — Appli- 
cation of Spherical Trigonometry. 

PREFACE 

Although this book has been designed to meet the specific 
needs of the Carnegie Technical Schools, the growing demand 
in the field of technical education for a form of mathematical 
instruction that will eliminate the purely speculative and 
concentrate the more utilitarian features of mathematical 
science, leads the author to believe that such a work as this 
will not be entirely inapt outside of the Carnegie Technical 
Schools.— S. S. K. 



#eonietrp 

Carnegie Cecj^ntcal ^c!)oolg 

8vo. Half Leather. 212 Pages. Price, $1.50 Net. 

CONTENTS 

PLANE : Definitions — Axioms — Geometrical Pro- 
cesses — Postulates — Perpendicular and Oblique Lines — 
Triangles — Quadrilateral — Polygons — Circle — Problems 
— Applications of Proportion — Areas — Regular Polygons 
and Circles. SOLID : Definitions — Geometrical Solids 
— Prisms and Parallelopipeds — Schematic Arrangement 
— Cylinders — Cones — Sphere. 

PREFACE 

It has been the author's endeavor in writing this book on 
Geometry to put the student at ease as far as possible by 
appealing from the start to his common sense. 

Technicalities have been avoided wherever possible and the 
student encouraged to think about the propositions in the 
same, simple, common sense way that he would consider any 
practical question arising in his daily experience. 

This process is applied even to the construction of figures, 
his ingenuity being called upon for suggestions as to auxiliary 
lines, etc. In this way, the solution of original propositions 
is gradually approached, until confidence is acquired. 

S. S. K. 



ilnalptical (^eomrtrp 
anb Calculus 

Carnegie Cec|)ntcal ^ctioolg 

PROF. S. S. KELLER and PROF. W. F. KNOX 

8vo. Half Leather. 359 Pages. Price, $2.00 Net. 



CONTENTS 

ANALYTICALGEOMETRY— Position— Dimensions— In- 
tersections — Miscellaneous Curves — Loci — The Straight Line — 
Transformation of Co-ordinates — Circle — Tangents and Nor- 
mals — Conic Sections — The Parabola. — The Ellipse — Supple- 
mental Chords — The Hyperbola — Asymptotes — Higher Plane 
Curves— The Cycloid— Roulettes— Spirals. CALCULUS— 
Fundamental Principles — Differentiation — Differentiation of 
Logarithmic and Exponential Functions — Integration — Tan- 
gents, Subtangents, Normals and Subnormals — Successive Dif- 
ferentiations—Evolution of Indeterminate Forms— Maxima and 
Minima — Partial Derivatives — Derivatives of Arcs, Areas, 
Volumes, etc. — Direction of Bending Curvature — Integration 
as a Summation — Integration by Parts — Trigonometric 
Integrals. 



elements! of 
$lane ^eometrp 

By Charles N. Schmall and Samuel M. Shack 
i2mo. Half Leather. 233 Pages. Illustrated. Price, ^1.25 Net. 

Cl^e M^tinttiU featurejs of tW 5^orfi are: 

^ I. The brevity, simplicity, and clearness of the 
demonstrations. Old and time-worn proofs have 
been replaced by new and more elegant solutions. 
^ 2. Exercises are given directly with most of the 
propositions. These help to fix firmly in mind the 
knowledge gained, and to stimulate the interest of the 
student by affording him a ready means of applying 
his knowledge. The exercises are abundant and 
carefully graded. 

^ 3. A thorough discussion is given on the methods 
of attacking original exercises, illustrated by examples. 
^ 4. The work has been so framed as to discourage 
the detrimental habit, on the part of pupils, of mem- 
orizing demonstrations. 

^ 5. References and details of demonstrations are 
gradually omitted. This promotes independence and 
also exercises the memory of the student in recollect- 
ing propositions. 

^ 6. The fundamental theorem of Limits is clearly 
explained. 

^ 7. A careful treatment has been given to the sub- 
ject of Maxima and Minima, and several new prop- 
ositions added. 



A FIRST COURSE IN 

9[Mlj>tical #eontetrp 

Pane anti ^oliti toitf) 
jBtutnetous; CjcamplejS 

By CHARLES N. SCHMALL 

Author of '^ Elements of Plane Geometry " 
i2mo. 318 Pages. Half Leather. Price, I1.75 Net. 



THE present work is designed as a textbook 
for colleges and scientific schools and is 
well adapted to the needs of these institu- 
tions. The chief aim is to give an easy 
and gradual development of the principles 
of the subject and to render it as attractive and inter- 
esting to the beginner as the nature of Analytical 
Geometry will allow. 

^ Among the many new features of this book the 
following are noteworthy. 

^ (i) The easy development of the fundamental re- 
lation between an equation and its locus, and vice 
versa. The first three chapters form a natural trans- 
ition from Elementary Geometry and pave the way 
for the rest of the book. 

^ (2) The use of the determinant notation wherever 
profitable. 

fl (3) The treatment of the equation of the second 
degree representing a pair of straight lines. 



^ (4) The development of the more elegant proper- 
ties of circles, such as the angle of intersection, coaxial 
circles, similitude, etc. 

^ (5) The treatment of the conic sections. Use of 
the eccentric angle, etc. The equations of the conic 
sections are derived from a single definition. 

^ (6) The chapter on the elementary properties of 
confocal conies. 

^ (7) The proof of the formulas for the area of a 
triangle and of a polygon in terms of the co-ordinates 
of the verities. 

^ (8) The numerous original exercises. Among 
them are some of the most elegant properties of the 
conies. The diagrams have been made as suggestive 
as desirable in order to help the student in his w^ork. 

^ It is believed that the author has succeeded in pro- 
ducing a work that is thoroughly modern, scientific, 
and up to date in every respect. 



A BRIEF COURSE IN THE 

Calculus 

By WILLIAM CAIN 

Professor of Mathematics in the University of North Carolina 
280 + 7 Pages, 63 Illustrations, Half Leather, 4 x 5 $^.^S Net. 

CONTENTS 

Graphs — Limits — Derivatives — Rates — Tangent and Nor- 
mal — Derivatives of Transcendental Functions — Higher De- 
rivatives. Derived Curves — Application to Mechanics — 
Infinite Series. Taylor's Theorem — McLaurin's Theorem. 
Expansion — Maxima and Minima. Concavity — Differentials. 
Derivatives of Area and Volume — Integration as anti- differ- 
entiation — Integration as a Limit of a Sum — Length of Arcs 
— Areas — Volumes — Centre of Mass — Moment of Inertia — 
Indeterminate Forms — Review Exercises. 



THE book presupposes some knowledge of 
Geometry, a working knowledge of Algebra 
through logarithms, and a thorough knowledge 
of the elements of Trigonometry. Two in- 
troductory chapters on Graphs will supply the 
student with all the actual needs of Analytic Geometry to 
read the book, but it is desirable, if possible that a brief 
course in Analytic Geometry should be studied before 
taking up the study of the Calculus. 



Botajger'g 



SERIES OF 






AMONG the Institutions in which the different 
works in this Series of Mathematics have 
been introduced and are now in use, are the 
following : 
Worcester Polytechnic Institute — ^Wesleyan University 
— University of Pennsylvania — ^Yale College — University 
of Georgia — Rutgers College — Mass. Polytechnic Institute 
— -University of Wisconsin — Iowa State University — 
Racine College — Ohio University — University of Texas 
— Syracuse University — Rennselaer Polytechnic Institute 
— Bowdoin College — Iowa College — Washington Univer- 
sity — Stevens Institute — Penn. Military Academy — 
Princeton University — U. S. Naval Academy — Lafayette 
College — Parsons College — Kansas State University — 
Chaddock College — Adrian College — Beloit College — 
Butler University — Delaware College — Furman Univer- 
sity — Tufts College. 



AN ELEMENTARY TREATISE ON 

^nalptic #eometrj> 

EMBRACING 

Pane (^tamttr^ 

AND AN 

S^ntroliurtion to 45eometrp of €!)rec aDtmcn^ionjBf 

TWENTY-SECOND EDITION 

i2mo., Cloth. 307 Pages. 117 Figures. Price, I1.75 

Contents: The Point — The Right Line — Trans- 
formation of Co-ordinates — The Circle — The Para- 
bola — The Ellipse — The Hyperbola — General Equa- 
tion of the Second Degree — Higher Plane Curves — 
The Plane — Surfaces of Revolution. 

THE present work on Analytic Geometry is 
designed as a textbook for Colleges and 
Scientific Schools. The object has been 
to exhibit the subject in a clear and simple 
manner, especially for the use of begin- 
ners, and at the same time to include all that students 
usually require in the regular undergraduate course. 

It is thought that among the merits of this book 
are the presentations of the symmetrical and normal 
forms of the equations of the right line and of the 
plane, the equations of the ellipsoid and of the plane 
tangent to the ellipsoid, and the formulae for the dis- 
tances of a point from a line and from a plane. 



ELEMENTARYTREATISE ON THE 

Btfferenttal anb Integral 

Calculus 

TWENTY-SECOND EDITION 

i2mo., Cloth. 451 Pages. 59 Figs. Price, $2.25 

Contents: First Principles — Differentiation of Algebraic 
and Transcendental Functions — Trigonometric Functions — 
Circular Functions — Limit — Derived Functions — Successive 
Differentials and Derivatives — Development of Functions 
— Evaluation of Indeterminate Form_s — Functions of Two or 
More Variables — Change of the Independent Variable — 
Maxima and Minima of Functions of a Single Variable 
— Tangents, Normals and Asymptoles^ — Direction of Curva- 
ture — Singular Points — Tracing of Curves — Radius of Cur- 
vature, Evolutes and Involutes — Envelopes — Elementary 
Forms of Integration — Integration of Rational Fractions — 
Integration of Irrational Functions b}^ Rationalization — 
Integration b}^ Successive Reductions — Integration by Series 
— Successive Integration — Integration of Functions of Two 
Variables — Definite Integrals — Length of Curves — Areas of 
Plane Curves — Areas of Curved Surfaces — Volumes of Solids. 



THE writer has adopted the method of infinitesimals, 
having learned from experience that the funda- 
mental principles of the subject are made more 
intelhgible to beginners by the method of in- 
finitesimals than by that of limits, while in the 
practical applications of the Calculus the inves- 
tigations are carried on entirely by the m.ethod of infinitesimals. 
At the same time, a thorough knowledge of the subject re- 
quires that the student should become acquainted with both 
methods; and for this reason, Chapter III is devoted ex- 
clusively to the method of limits. In this chapter all the 
fundamental rules for differentiating algebraic and trans- 
cendental functions are obtained by the method of limits, so 
that the student may compare the two methods. 



AN ELEMENTARY TREATISE ON 






SEVENTEENTH EDITION 



i2mo., Cloth. 511 Pages. 102 Figures. Price, S3. 00 

Contents: First Principles — Statics (Rest) — Cen- 
tre of Gravity — (Centre of Mass) — Friction — Virtual 
Velocities — Machines — Funicular Polygon — Catenary 
— Attraction — Rectilinear Motion — Curvilinear Mo- 
tion — Lav^s of Motion — Central Forces — Constrained 
Motion — Impact — Work and Energy — Moment of 
Inertia — Rotary Motion — Motion of a System of 
Rigid Bodies in Space. 



THE book consists of three parts. Part I, 
with the exception of a preliminary chap- 
ter devoted to definitions and fundamen- 
tal principles, is entirely given to Statics. 
Part II is occupied with Kinematics, and 
the principles of this important branch of mathematics 
are so treated that the student may enter upon the 
study of Kinetics with clear notions of motion, ve- 
locity and acceleration. Part III treats of the 
Kinetics of a particle and of rigid bodies. 




AN ELEMENTARY TREATISE ON 

ptiromecJjanicsi 

FIFTH EDITION 
i2mo.. Cloth. 298 Pages. 85 Figs. Price, I2.50 

Contents: Equilibrium and Pressure of Fluids — 
Equilibrium of Floating Bodies — Specific Gravity — 
Equilibrium and Pressure of Gases — Elastic Fluids — 
Motion of Liquids — Efflux — Resistance and Work 
of Liquids — Motion of Water in Pipes and Open 
Channels — Motion of Elastic Fluids — Hydrostatic 
and Hydraulic Machines. 



THE present work on Hydromechanics is 
designed as a text-book for Scientific 
Schools and Colleges, and is prepared on 
the same general plan as the author's Ana- 
lytic Mechanics, which it is intended to 
follow. Like the Analytic Mechanics, it involves the 
use of Analytic Geometry and the Calculus, though a 
geometric proof has been introduced wherever it 
seemed preferable. 

The book is divided into two parts, namely, Hy- 
drostatics and Hydrokinetics. The former is subdi- 
vided into three, and the latter into four chapters; 
and at the ends of the chapters a large number of ex- 
amples is given, with a view to illustrate every part of 
the subject. Many of these examples were prepared 
specially for this work, and are practical questions in 
hydraulics, etc., taken from every-day life. 



A TREATISE ON 

Witt^ 0umtxom €xtxtmsi 

i2mo., Cloth. 205 Pages, Illustrated. Price, $2.25 Net. 

PARTIAL TABLE OF CONTENTS 

Definitions — The Dead Load — The Live Load — The Apex 
Loads and Reactions — Relations between External Forces 
and Internal Stresses — Methods of Calculation — Lever Arms — 
Indeterminate Cases — Snow Load Stresses — Wind Loads — 
Complete Calculation of a Roof Truss — Different Forms of 
Trusses — Shear — Shearing Stress — Web Stresses due to Dead 
Loads — Chord Stresses due to Dead Loads — Position of 
Uniform Live Load causing Maximum Chord Stresses — 
Maximum Stresses in the Chords — Position of Uniform Live 
Load causing Maximum Shears — The Warren Truss — Mains 
and Counters— The Howe Truss— The Pratt Truss— The 
Warren Truss with Vertical Suspenders — The Double Warren 
Truss— The Whipple Truss— The Lattice Truss— The Para- 
bolic Bowstring Truss — The Circular Bowstring Truss — Snow 
Load Stresses — Stresses due to Wind Pressure — The Factor of 
Safety. Preliminary Statement — When the Uniform Train 
Load is preceded by One or More Heavy Excess Panel Loads — 
When One Concentrated Excess Load accompanies a Uniform 
Train Load — When Two Equal Concentrated Excess Loads 
accompany a Uniform Train Load — The Baltimore Truss — 
The Maximum Shears for Uniform Live Load — Locomotive 
Wheel Loads — Position of Wheel Loads for Maximum Shear — 
Position of Wheel Loads for Maximum Moment at Joint in 
Loaded Chord — In Unloaded Chord — Tabulation of Moments 
of Wheel Loads. 



Sntefiratton hp Crtgonometric 

anti 

Jfmagtnarp Substitution 

By PROF. C. O. GUNTHER 

With an Introduction by 

Prof. J. BURKITT WEBB 

(Stevens' Institute) 
i2mo.5 Cloth. 76 Pages. Illustrated. 1 1.25 Net. 



A METHOD of integration, which may be 
called the ''Triangle Method/' has been 
used successfully for the past few years 
by the author in his classes. The primary 
object in view has been to eliminate the 
^^ Reduction Formulae/^ and make the student inde- 
pendent of text-books and tables of integrals. This 
method is formed upon trigonometric principles with 
the result that the student gains proficiency not only 
in the integration of trigonometric differential ex- 
pressions but also in the transformation of algebraic 
expressions into trigonometric and exponential ones, 
and vice versa. 

^ This book is intended to be used in conjunction 
with the usual text books on the subject, and should 
be taken up after the student has become familiar 
with the simple rules of integration, resulting from 
the reversion of the rules for differentiation. — Preface. 



Cl)amb0rs' 

3togaritf)m^ of l^umBer^, X to 108000 
Crigonometncal anti i^autical ^abW 

Edited by JAMES PRYDE, F.E.LS. 
8vo., Cloth. 454 Pages. NEW EDITION. Price, ^1.75 



PART— CONTENTS 

AMPLITUDES— Angles Contained by the 
Meridian with every Point and Quarter- 
Point of the Compass — Apparent Depres- 
sion of the Horizon — Areas, Apothems, 
and Angles of Polygons — Augmentation 
of the Noon's Semi-diameter — Bessel's Refractions — 
Binomial Coefficients for Interpolation by Differ- 
ences — Circular Measure of Angles — Contraction of 
the Semi-diameters of the Sun and Moon — Correc- 
tion of the Moon's Meridian Passage — Diurnal Pro- 
portional Logarithms — Equation of Equal Altitudes 
— Height of Apparent, above True Level — Mean 
Motion of Sun for Periods of Mean Time — Mean Re- 
fraction — Meridional Parts — Mutual Conversion of 
Sidereal and Mean Time — Quarter Squares Nos. i 
to 5100 — Reduction of Common to Naperian Loga- 
rithms — Reduction of the Moon's Equatorial Paral- 
lax — Reduction of the Moon's Horizontal Parallax 
and semi-diameter to any given Greenwich Date — 
Semi-diurnal and Semi-nocturnal Arcs — Sun's Par- 
allax in Altitude — Ternary Proportional Logarithms 
— Traverse Table, or Difference of Latitude and 
Departure. 



practical J^atfjematics 

REVISED UNDER SUPERVISION 
OF 

C. G. KNOTT, D.Sc, F.R.S.E. 

AND 
J. S. MACKAY, M.A., LL.D. 

8vo., Cloth. 627 Pages. Illustrated. Price, ^2.00 
WITH MANY FIGURES, DIAGRAMS AND TABLES. 

Contents : Descriptive Geometry — Computation 
by Logarithms — Log. Scales — The Lines of the Sec- 
tor — Plane Trigonometry — Mensuration of Heights 
and Distances — Mensuration of Surfaces — Land Sur- 
veying — Mensuration of Solids — Mensuration of Conic 
Sections — Solids of Revolution of the Conic Sections — 
Regular Solids — Cylindric Rings — Spindles — Ungu- 
las — Irregular Solids — The Common Sliding Rule — 
Measurement of Timber — Relations of Weight and 
Volume of Bodies — Arched Roofs — Gauging — Baro- 
metric Measurement of Heights — Measurement of 
Distances by the Velocity of Sound — Measurement 
of Heights and Distances — Levelling — Strength of 
Materials and their Essential Properties — Projectiles 
and Gunnery — Projections — Stereographic Projec- 
tion of the Cases of Trigonometry — Spherical 
Trigonometry — Astronomical Problems — Navigation 
— Construction of Maps and Charts — Geodetic Sur- 
veying — Curve Tracing — Tables — Four-Place Loga- 
rithms of Numbers and Circular Functions. 



Biffermtial Equations 

of tfjc 

jFirst Species 



By WILLIAM J. BERRY, C.E., M.S. 

Instructor in Mathematics in the Polytechnic Institute of Brooklyn 



i2mo., Cloth. Illustrated. In Press. 



THIS book is intended for engineers and 
students of engineering who desire to be- 
come proficient in the use of differential 
equations without spending too much time 
in study. To this end the equations are 
grouped according to certain standard forms, the 
solutions of which are carefully developed, and sum- 
marized in a plainly stated rule. Under each type, 
illustrative examples are given and fully worked out, 
no essential step being omitted. Numerous other 
examples are supplied to be solved by the student. 
While the proofs are mathematically rigid, the aim 
has been to make the presentation so simple that any- 
one with an elementary knowledge of the calculus 
can understand the reasoning and apply the rules. 
There is included a table of the forms with their 
rules, and a table of the more commonly used inte- 
grals. The chief points of the book are its Clearness, 
Conciseness, Convenience. 



atiti practical Applications 



By Wm. S. hall, C.E., E.M., M.S. 

Professor of Graphics and Mining in Lafayette College^ Easton^ Pa, 

WWi^ a ^mwctti atlais of is piatejs 

8vo., Cloth. 76 Pages. Price, I3.50 Net. 



CONTENTS 

FIRST PRINCIPLES: Definitions— Projections ot 
a Point — Revolutions of Vertical Plane^Projections of a 
Line — Representation of Planes — Notation and Conven- 
tions—Profile Planes. PROBLEMS ON THE POINT, 
LINE AND PLANE: Traces of a Line— Length of a 
Line — Points and Lines in Oblique Planes — Revolution 
of Points and Lines — Plane Angles — Planes — Additional 
Problems for Construction. CURVES AND TANGENTS 
TO CURVES: Generation and Classification of Lines — 
Curves and Tangents. SURFACES: Generation and 
Classification of Surfaces — Single Curved Surfaces and 
Tangent Planes — Surfaces of Revolution — Intersections 
and Developments — ^Warped Surfaces — Additional Prob- 
lems on Surfaces. 



ELEMENTS OF THE 

Biffermtial anlj Snttgral 

Calculus 

By Wm. S. hall, C.E., E.M., M.S. 

Professor of Technical Mathematics in Lafayette College, 
SECOND EDITION : 8vo., Cloth. 249 Pages. Price, $z,z^ Net. 

CONTENTS 

Definitions and First Principles — Differentiation of 
Algebraic Functions — Differentiation of Transcendental 
Functions — Differentials — Integration — Successive Differ- 
entiation and Integration — Applications in Mechanics — 
Functions of Two or More Variables — Implicit Functions 
— Change of the Independent Variable — Development 
of Functions — Evaluation of Indeterminate Forms — Max- 
ima and Minima of Functions — Tangents, Normals and 
Asymptotes — Direction of Curvature — E volutes and Invo- 
lutes — Envelopes — Singular Points — Integration of Ration- 
al Fractions — of Irrational — Integration by Parts and by 
Successive Reduction — Integration by Transcendental 
Functions — by Series — as a Summation — Areas and 
Lengths of Plane Curves — Surfaces and Volumes of 
Solids — Centre of Mass — Moment of Inertia — Properties 
of Guldin — Differential Equations — Appendix. 



Hogaritljmic Cables; 

of jBumljers; anti 

Crtgonometrtcal JTuncttons; 

By baron von VEGA 

Translated from the Fortieth or Dr. Bremiker's thoroughly revised 

and enlarged edition by 

W. L. F. FISCHER, M.A., F.R.S. 

Felloiv of Clare College, Cambridge, 
Professor of Natural Philosophy, Uni'versity of St, Andreiv. 

8vo., Cloth. 575 Pages. Price, $2.^^ 



THE existing logarithmic-trigonometrical tables 
to seven figures may be divided into three 
classes. While the first class which contains 
the logarithms of the natural numbers is 
very nearly the same in all, they diflTer in the 
second, or trigonometrical part of this, that the first class 
contains the trigonometrical functions for the greatest 
part of the quadrant only for every full minute, the second 
for every tenth second, and the third for every single 
second. Among the improvements in these tables we 
may mention: The systematic arrangement of all the 
pages of each section, which has for its object that, when 
once the book is opened at the proper page, the eye will 
involuntarily be directed to the place where the required 
logarithm is to be found; The legible type; The addition 
of small tables of proportional parts in the first and third 
parts, to facilitate the interpolation; and the special 
attention paid to the correctness of the seventh decimal 
place. 



A NEW MANUAL OF 



TO 



^eben plates of Becimate 

Edited by DR. BRUHNS 

Director of the Obser-vatory^ and Professor of Astronomy at Leipzig, 

SEVENTH EDITION 



8vo., Cloth. 6io Pages. Price, I2.50 

THE logarithms of the numbers from i to 
108000 have been reduced to the extent of 
from I to looooo since the addition from 
1 00000 to 108000 does not appear to offer a 
sufficient advantage. The logarithms of the 
first 6 degrees of the trigonometric functions, Sine, Cosine, 
Tangent and Cotangent have been given to every second, 
with the addition of the differences and where the space 
would allow of it, of the proportional parts. Such ar- 
rangement will save much labor spent in interpolation. 
The remaining 39 degrees of the trigonometrical functions 
are given to every 10 seconds. The present work with the 
exception of some few small additional tables consists 
merely of the logarithms of numbers and of the trigono- 
metrical logarithms. 

The arrangement of the tables is on a plan easily under- 
stood, not overladen with instructions, printed with a 
clear distinct type and of undoubted correctness. 



.**..ti^:-.V co*.c:^.% >*\-J4;-.V .0* 














